Graphics messing up array alignment?
You probably need a longtable and to make the matrices smaller; I also suggest to split the longer polynomials over two lines.
\documentclass[12pt,letterpaper]{report}
\usepackage{amsmath}
\usepackage{graphicx}
\usepackage[export]{adjustbox}
\usepackage{booktabs,longtable,array}
\newenvironment{smallbmatrix}
{\left[\smallmatrix}
{\endsmallmatrix\right]}
\begin{document}
\begin{small}
\begin{longtable}{@{}*{4}{>{$}c<{$}}@{}}
\caption{Knots}\label{tab:knots} \\
\midrule[\heavyrulewidth]
\textbf{Knot} &
\textbf{%
\begin{tabular}{@{}c@{}}
Knot with \\
Crossings Labeled \\
and Oriented
\end{tabular}%
} &
\textbf{Trip Matrix} &
\multicolumn{1}{c}{\textbf{Jones Polynomial}}
\\
\midrule
\endfirsthead
\caption*{Knots (continued} \\
\midrule[\heavyrulewidth]
\textbf{Knot} &
\textbf{%
\begin{tabular}{@{}c@{}}
Knot with \\
Crossings Labeled \\
and Oriented
\end{tabular}%
} &
\textbf{Trip Matrix} &
\multicolumn{1}{c}{\textbf{Jones Polynomial}}
\\
\midrule
\endhead
\bottomrule
\endfoot
8_{18}
&
\includegraphics[valign=c,width=0.25\textwidth]{example-image.jpg}
&
\begin{smallbmatrix}
0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\
1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\
1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 & 0
\end{smallbmatrix}
&
\begin{aligned}
q^4 & -4q^3+6q^2-7q+9 \\
& -7q^{-1}+6q^{-2}-4q^{-3}+q^{-4}
\end{aligned}
\\
\addlinespace
8_{19}
&
\includegraphics[valign=c,width=0.25\textwidth]{example-image.jpg}
&
\begin{smallbmatrix}
0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \\
1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 & 0 & 1 & 1 & 0
\end{smallbmatrix}
&
-q^8+q^5+q^3
\\
\addlinespace
8_{20}
&
\includegraphics[valign=c,width=0.25\textwidth]{example-image.jpg}
&
\begin{smallbmatrix}
1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\
1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\
1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\
\end{smallbmatrix}
&
\begin{aligned}
-q & +2-q^{-1}+2q^{-2}\\
& -q^{-3}+q^{-4}-q^{-5}
\end{aligned}
\\
\addlinespace
8_{21}
&
\includegraphics[valign=c,width=0.25\textwidth]{example-image.jpg}
&
\begin{smallbmatrix}
0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\
1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\
\end{smallbmatrix}
&
\begin{aligned}
2q^{-1} & -2q^{-2}+3q^{-3}-3q^{-4} \\
& +2q^{-5}-2q^{-6}+q^{-7}
\end{aligned}
\\
\end{longtable}
\end{small}
\end{document}
- your tale is to wide to be fit on page
- with use
mathtoolspackage is possible break long math expression oin the last column in two lines - for vertical adjustment images in table i suggest to use
adjustboxpackage - for more vertical space around matrices i suggest to use macro
\makegapedcellsfrommakecellpackage:
\documentclass[12pt,letterpaper]{report}
\usepackage[export]{adjustbox} % <--- changed
\usepackage{mathtools} % <--- changed
\usepackage{makecell}
\renewcommand\theadfont{\bfseries}
\renewcommand\theadgape{}
\setcellgapes{3pt}
\begin{document}
\begingroup\small
\makegapedcells
\[
\begin{array}{cccc}
\text{\thead[b]{Knot}}
& \text{\thead[b]{Knot with Crossings\\ Labeled and Oriented}}
& \text{\thead[b]{Trip Matrix}}
& \text{\thead{Jones Polynomial}} \\
\text{\Large $8_{18}$}
& \includegraphics[width=0.25\textwidth, valign=c]{example-image.jpg}
& \begin{bmatrix}
0 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\
1 & 0 & 1 & 0 & 1 & 0 & 0 & 1 \\
0 & 1 & 0 & 1 & 1 & 1 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & 1 & 1 & 0 \\
0 & 1 & 1 & 0 & 0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 & 1 & 0 & 1 & 0 \\
1 & 0 & 0 & 1 & 0 & 1 & 0 & 1 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 & 0
\end{bmatrix}
& \begin{multlined}
q^4 - 4q^3 + 6q^2 - 7q + 9 - \\
7q^{-1} +6q^{-2}-4q^{-3}+q^{-4}
\end{multlined} \\
\text{\Large{$8_{19}$}}
& \includegraphics[width=0.25\textwidth,valign=c]{example-image.jpg}
& \begin{bmatrix}
0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \\
1 & 0 & 1 & 0 & 1 & 1 & 0 & 0 \\
1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 \\
0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 \\
0 & 0 & 1 & 1 & 0 & 1 & 1 & 0
\end{bmatrix}
& -q^8+q^5+q^3 \\
\text{\Large{$8_{20}$}}
& \includegraphics[width=0.25\textwidth,valign=c]{example-image.jpg}
& \begin{bmatrix}
1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\
1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\
1 & 0 & 0 & 1 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 & 1 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\
\end{bmatrix}
& \begin{multlined}
-q+2 - q^{-1} + 2q^{-2} - \\
q^{-3} + q^{-4} -q^{-5}
\end{multlined} \\
\text{\Large{$8_{21}$}}
& \includegraphics[width=0.25\textwidth,valign=c]{example-image.jpg}
& \begin{bmatrix}
0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\
1 & 1 & 0 & 1 & 0 & 0 & 1 & 1 \\
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \\
0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\
1 & 0 & 1 & 1 & 1 & 1 & 0 & 0 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\
1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\
\end{bmatrix}
& \begin{multlined}
2q^{-1} - 2q^{-2} + 3q^{-3} - \\
3q^{-4} + 2q^{-5} - 2q^{-6} + q^{-7}
\end{multlined}
\end{array}
\]
\endgroup
\end{document}
(red lines indicate your page layout)
Ad 1.
A macro to center the argument vertically around the math axis:
\newcommand{\vchbox}[1]{\ensuremath{\vcenter{\hbox{#1}}}}
Usage:
\vchbox{\includegraphics[...]{..}}