8 1

7_7

8_2

Visit 8 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 1's page at Knotilus! Visit 8 1's page at the original Knot Atlas! 8 1 Quick Notes

8 1 Further Notes and Views

Knot presentations

Planar diagram presentation	X1425 X9,12,10,13 X3,11,4,10 X11,3,12,2 X5,16,6,1 X7,14,8,15 X13,8,14,9 X15,6,16,7
Gauss code	-1, 4, -3, 1, -5, 8, -6, 7, -2, 3, -4, 2, -7, 6, -8, 5
Dowker-Thistlethwaite code	4 10 16 14 12 2 8 6
Conway Notation	[62]

Minimum Braid Representative: Length is 10, width is 5. Braid index is 5.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index $\{4,5\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-7][-3] Hyperbolic Volume 3.42721 A-Polynomial See Data:8 1/A-polynomial

[edit Notes for 8 1's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $1$ Rasmussen s-Invariant 0

[edit Notes for 8 1's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t+7-3t^{-1}$
Conway polynomial	$1-3z^{2}$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 13, 0 }
Jones polynomial	$q^{2}-q+2-2q^{-1}+2q^{-2}-2q^{-3}+q^{-4}-q^{-5}+q^{-6}$
HOMFLY-PT polynomial (db, data sources)	$a^{6}-z^{2}a^{4}-a^{4}-z^{2}a^{2}-z^{2}+a^{-2}$
Kauffman polynomial (db, data sources)	$a^{5}z^{7}+a^{3}z^{7}+a^{6}z^{6}+2a^{4}z^{6}+a^{2}z^{6}-5a^{5}z^{5}-4a^{3}z^{5}+az^{5}-5a^{6}z^{4}-8a^{4}z^{4}-2a^{2}z^{4}+z^{4}+7a^{5}z^{3}+5a^{3}z^{3}-az^{3}+z^{3}a^{-1}+6a^{6}z^{2}+7a^{4}z^{2}+z^{2}a^{-2}-3a^{5}z-3a^{3}z-a^{6}-a^{4}-a^{-2}$
The A2 invariant	$q^{20}+q^{18}-q^{12}-q^{10}+q^{-2}+q^{-6}+q^{-8}$
The G2 invariant	$q^{94}+q^{90}-q^{88}+2q^{80}-2q^{78}+q^{76}+q^{74}+q^{70}-q^{68}+q^{64}+q^{54}-q^{52}-q^{46}-q^{42}+q^{40}-q^{38}+q^{36}-q^{34}-2q^{32}+q^{30}-q^{28}-q^{22}+q^{18}-q^{12}+q^{8}+q^{-2}-q^{-6}+q^{-10}+q^{-14}+q^{-20}+q^{-24}+q^{-28}+q^{-34}+q^{-38}$

Further Quantum Invariants

Further quantum knot invariants for 8_1.

A1 Invariants.

Weight	Invariant
1	$q^{13}-q^{7}+q^{-1}+q^{-5}$
2	$q^{38}-q^{34}-q^{28}+q^{24}+q^{12}+q^{10}-1+q^{-8}+q^{-14}$
3	$q^{75}-q^{71}-q^{69}+q^{65}-q^{61}+q^{57}+q^{55}-q^{51}+q^{37}+q^{35}-q^{31}-q^{25}-q^{23}-q^{17}+q^{13}+2q^{11}+q^{9}+q^{7}-q^{5}+q+q^{-1}-q^{-5}+q^{-9}-q^{-13}-q^{-15}+q^{-19}+q^{-27}$
4	$q^{124}-q^{120}-q^{118}-q^{116}+q^{114}+q^{112}+q^{110}-2q^{106}+q^{102}+q^{100}+q^{98}-q^{96}-q^{94}-q^{92}+q^{88}+q^{74}+q^{72}-q^{68}-2q^{66}+q^{62}-q^{58}-2q^{56}+2q^{52}+q^{50}-q^{46}+2q^{42}+q^{40}+q^{32}-q^{28}-q^{26}+q^{22}-q^{18}-q^{16}-q^{14}-q^{12}+q^{10}+2q^{8}+q^{6}-q^{4}-2q^{2}+2q^{-2}+4q^{-4}+q^{-6}-2q^{-8}-q^{-10}+q^{-12}+3q^{-14}-2q^{-18}-q^{-20}+2q^{-24}-q^{-28}-q^{-30}-q^{-32}+q^{-34}+q^{-44}$
5	$q^{185}-q^{181}-q^{179}-q^{177}+q^{173}+2q^{171}+q^{169}-q^{165}-2q^{163}-q^{161}+q^{159}+2q^{157}+q^{155}-q^{151}-2q^{149}-q^{147}+q^{143}+q^{141}+q^{139}-q^{135}+q^{123}+q^{121}-q^{117}-2q^{115}-2q^{113}+2q^{109}+2q^{107}-2q^{103}-2q^{101}-q^{99}+2q^{97}+4q^{95}+2q^{93}-q^{91}-2q^{89}-2q^{87}+2q^{83}+2q^{81}-2q^{77}-2q^{75}+q^{71}+2q^{69}+q^{67}-q^{65}-2q^{63}-q^{61}+q^{57}+q^{55}-q^{53}-2q^{51}-q^{49}+q^{45}-q^{41}+q^{37}+3q^{35}+2q^{33}-q^{25}+q^{23}+q^{21}+q^{19}+q^{17}-3q^{13}-3q^{11}-q^{9}+q^{7}+3q^{5}+2q^{3}-3q^{-1}-3q^{-3}+3q^{-7}+3q^{-9}-2q^{-13}-2q^{-15}+3q^{-19}+2q^{-21}-2q^{-25}+2q^{-29}+2q^{-31}+q^{-33}-q^{-35}-2q^{-37}-q^{-39}+q^{-41}+q^{-43}-q^{-49}-q^{-51}+q^{-65}$
6	$q^{258}-q^{254}-q^{252}-q^{250}+2q^{244}+2q^{242}+q^{240}-q^{236}-2q^{234}-3q^{232}+q^{228}+2q^{226}+2q^{224}+q^{222}-3q^{218}-2q^{216}-q^{214}+q^{210}+2q^{208}+2q^{206}-q^{200}-q^{198}-q^{196}+q^{192}+q^{184}+q^{182}-q^{178}-2q^{176}-2q^{174}-2q^{172}+q^{170}+3q^{168}+3q^{166}+2q^{164}-q^{162}-3q^{160}-4q^{158}-q^{156}+2q^{154}+4q^{152}+5q^{150}+2q^{148}-2q^{146}-5q^{144}-4q^{142}-2q^{140}+q^{138}+4q^{136}+3q^{134}+q^{132}-2q^{130}-3q^{128}-3q^{126}-q^{124}+3q^{122}+3q^{120}+3q^{118}+q^{116}-q^{114}-3q^{112}-4q^{110}-q^{108}+q^{106}+2q^{104}+2q^{102}+q^{100}-2q^{98}-4q^{96}-2q^{94}+q^{92}+3q^{90}+3q^{88}+2q^{86}-q^{84}-4q^{82}-q^{80}+2q^{78}+3q^{76}+3q^{74}+2q^{72}-q^{70}-3q^{68}-q^{66}+q^{64}+q^{62}-q^{58}-2q^{56}-2q^{54}+q^{50}-q^{46}-2q^{44}-2q^{42}-q^{40}+q^{38}+2q^{36}+3q^{34}+2q^{32}+q^{30}-q^{26}-3q^{24}-3q^{22}+3q^{18}+4q^{16}+4q^{14}+3q^{12}-2q^{10}-4q^{8}-4q^{6}-q^{4}+2q^{2}+5+6q^{-2}+q^{-4}-2q^{-6}-5q^{-8}-4q^{-10}-2q^{-12}+2q^{-14}+4q^{-16}+q^{-18}-2q^{-22}-q^{-24}-q^{-26}+q^{-30}-q^{-32}+q^{-34}+q^{-36}+2q^{-38}-q^{-42}-q^{-44}-2q^{-46}+q^{-48}+2q^{-50}+3q^{-52}+q^{-54}-q^{-58}-2q^{-60}+q^{-66}-q^{-74}-q^{-78}+q^{-90}$

A2 Invariants.

Weight	Invariant
1,0	$q^{20}+q^{18}-q^{12}-q^{10}+q^{-2}+q^{-6}+q^{-8}$
1,1	$q^{52}+2q^{48}-2q^{46}+2q^{44}-4q^{42}+2q^{40}-2q^{38}-2q^{32}+2q^{30}-3q^{28}+4q^{26}-2q^{24}+4q^{22}-2q^{20}+4q^{18}+2q^{14}+2q^{12}-2q^{6}-2q^{4}-2q^{2}-2+q^{-4}+2q^{-8}+2q^{-12}+2q^{-16}+q^{-20}$
2,0	$q^{52}+q^{50}+q^{48}-q^{46}-q^{44}-q^{42}-q^{40}-q^{38}-q^{36}+q^{34}+q^{32}+q^{30}+q^{18}+2q^{16}+q^{14}+q^{12}+q^{10}-q^{4}-2q^{2}-2-q^{-2}+q^{-4}+q^{-10}+q^{-12}+q^{-16}+q^{-18}+q^{-20}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{40}+q^{36}-q^{32}-q^{30}-q^{28}+q^{24}+2q^{22}+q^{20}+2q^{18}-q^{14}-q^{12}-q^{10}-q^{8}-q^{6}+q^{-4}+q^{-8}+2q^{-10}+q^{-12}+q^{-16}$
1,0,0	$q^{27}+q^{25}+q^{23}-q^{17}-q^{15}-q^{13}+q^{-3}+q^{-7}+q^{-9}+q^{-11}$

B2 Invariants.

Weight	Invariant
0,1	$q^{40}+q^{36}+q^{32}-q^{30}+q^{28}-q^{24}-q^{20}-2q^{16}+q^{14}-q^{12}+q^{10}-q^{8}+q^{6}+q^{-4}+q^{-8}+q^{-12}+q^{-16}$
1,0	$q^{66}+q^{58}-q^{54}-q^{52}-q^{46}-q^{44}+q^{40}+q^{38}+q^{36}+q^{32}+q^{30}+q^{28}-q^{18}-q^{16}-q^{10}-q^{8}+q^{-6}+q^{-14}+q^{-16}+q^{-18}+q^{-26}$

G2 Invariants.

Weight	Invariant
1,0	$q^{94}+q^{90}-q^{88}+2q^{80}-2q^{78}+q^{76}+q^{74}+q^{70}-q^{68}+q^{64}+q^{54}-q^{52}-q^{46}-q^{42}+q^{40}-q^{38}+q^{36}-q^{34}-2q^{32}+q^{30}-q^{28}-q^{22}+q^{18}-q^{12}+q^{8}+q^{-2}-q^{-6}+q^{-10}+q^{-14}+q^{-20}+q^{-24}+q^{-28}+q^{-34}+q^{-38}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["8 1"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-3t+7-3t^{-1}$

In[5]:=

Conway[K][z]

Out[5]=

$1-3z^{2}$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 13, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{2}-q+2-2q^{-1}+2q^{-2}-2q^{-3}+q^{-4}-q^{-5}+q^{-6}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$a^{6}-z^{2}a^{4}-a^{4}-z^{2}a^{2}-z^{2}+a^{-2}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$a^{5}z^{7}+a^{3}z^{7}+a^{6}z^{6}+2a^{4}z^{6}+a^{2}z^{6}-5a^{5}z^{5}-4a^{3}z^{5}+az^{5}-5a^{6}z^{4}-8a^{4}z^{4}-2a^{2}z^{4}+z^{4}+7a^{5}z^{3}+5a^{3}z^{3}-az^{3}+z^{3}a^{-1}+6a^{6}z^{2}+7a^{4}z^{2}+z^{2}a^{-2}-3a^{5}z-3a^{3}z-a^{6}-a^{4}-a^{-2}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {...}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n70, ...}

Vassiliev invariants

V2 and V3:

(-3, 3)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-12$ $24$ $72$ $66$ $38$ $-288$ $-432$ $-96$ $-104$ $-288$ $288$ $-792$ $-456$ ${\frac {1009}{10}}$ ${\frac {2626}{15}}$ $-{\frac {6182}{15}}$ ${\frac {943}{6}}$ $-{\frac {1551}{10}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $s=$0 is the signature of 8 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -6 -5 -4 -3 -2 -1 0 1 2 χ 5                 1 1 3                   0 1             2 1   1 -1           1 1     0 -3         1 1       0 -5       1 1         0 -7       1           -1 -9   1 1             0 -11                   0 -13 1                 1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-6$ ${\mathbb {Z} }$ $r=-5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{6}-q^{5}+2q^{3}-2q^{2}+2-3q^{-1}+q^{-2}+2q^{-3}-3q^{-4}+q^{-5}+3q^{-6}-3q^{-7}+3q^{-9}-3q^{-10}+3q^{-12}-2q^{-13}-q^{-14}+2q^{-15}-q^{-16}-q^{-17}+q^{-18}$
3	$q^{12}-q^{11}+2q^{8}-2q^{7}-q^{6}+3q^{4}-q^{3}-2q^{2}-q+4-2q^{-2}-2q^{-3}+3q^{-4}+2q^{-5}-2q^{-6}-q^{-7}+2q^{-8}+q^{-9}-3q^{-10}+2q^{-12}-3q^{-14}+q^{-15}+2q^{-16}-q^{-17}-2q^{-18}+2q^{-19}+2q^{-20}-2q^{-21}-2q^{-22}+2q^{-23}+2q^{-24}-2q^{-25}-2q^{-26}+q^{-27}+3q^{-28}-q^{-29}-2q^{-30}+2q^{-32}-q^{-34}-q^{-35}+q^{-36}$
4	$q^{20}-q^{19}+2q^{15}-3q^{14}+q^{11}+4q^{10}-5q^{9}-q^{8}-q^{7}+3q^{6}+7q^{5}-7q^{4}-3q^{3}-2q^{2}+6q+10-9q^{-1}-5q^{-2}-4q^{-3}+7q^{-4}+12q^{-5}-8q^{-6}-6q^{-7}-6q^{-8}+7q^{-9}+12q^{-10}-8q^{-11}-5q^{-12}-5q^{-13}+6q^{-14}+11q^{-15}-8q^{-16}-4q^{-17}-4q^{-18}+5q^{-19}+11q^{-20}-8q^{-21}-3q^{-22}-3q^{-23}+3q^{-24}+10q^{-25}-7q^{-26}-2q^{-27}-2q^{-28}+q^{-29}+8q^{-30}-6q^{-31}-q^{-32}-q^{-33}+6q^{-35}-5q^{-36}+5q^{-40}-5q^{-41}+5q^{-45}-4q^{-46}-q^{-47}-q^{-48}+5q^{-50}-2q^{-51}-q^{-52}-q^{-53}-q^{-54}+3q^{-55}-q^{-58}-q^{-59}+q^{-60}$
5	$q^{30}-q^{29}+q^{24}-2q^{23}+q^{21}+q^{19}+q^{18}-4q^{17}-q^{16}+2q^{15}+2q^{14}+2q^{13}+q^{12}-6q^{11}-3q^{10}+4q^{9}+4q^{8}+3q^{7}-2q^{6}-8q^{5}-3q^{4}+6q^{3}+7q^{2}+3q-5-11q^{-1}-3q^{-2}+9q^{-3}+9q^{-4}+4q^{-5}-7q^{-6}-13q^{-7}-5q^{-8}+9q^{-9}+12q^{-10}+5q^{-11}-7q^{-12}-13q^{-13}-5q^{-14}+7q^{-15}+13q^{-16}+5q^{-17}-7q^{-18}-11q^{-19}-4q^{-20}+5q^{-21}+12q^{-22}+4q^{-23}-6q^{-24}-10q^{-25}-5q^{-26}+4q^{-27}+11q^{-28}+5q^{-29}-4q^{-30}-10q^{-31}-6q^{-32}+3q^{-33}+10q^{-34}+6q^{-35}-2q^{-36}-9q^{-37}-7q^{-38}+2q^{-39}+8q^{-40}+6q^{-41}-7q^{-43}-7q^{-44}+6q^{-46}+6q^{-47}+q^{-48}-4q^{-49}-5q^{-50}-2q^{-51}+3q^{-52}+5q^{-53}+q^{-54}-2q^{-55}-3q^{-56}-2q^{-57}+q^{-58}+3q^{-59}+q^{-60}-q^{-61}-2q^{-62}-q^{-63}+q^{-64}+2q^{-65}+q^{-66}-q^{-67}-2q^{-68}-q^{-69}+3q^{-71}+2q^{-72}-q^{-73}-2q^{-74}-2q^{-75}-q^{-76}+2q^{-77}+3q^{-78}-q^{-80}-q^{-81}-2q^{-82}+2q^{-84}+q^{-85}-q^{-88}-q^{-89}+q^{-90}$
6	$q^{42}-q^{41}-q^{36}+2q^{35}-2q^{34}+q^{33}+q^{30}-2q^{29}+2q^{28}-4q^{27}+2q^{26}+q^{25}+q^{24}+3q^{23}-3q^{22}+q^{21}-7q^{20}+3q^{19}+q^{18}+2q^{17}+5q^{16}-4q^{15}+q^{14}-9q^{13}+5q^{12}+q^{10}+5q^{9}-5q^{8}+3q^{7}-8q^{6}+8q^{5}-2q^{4}-3q^{3}+3q^{2}-6q+6-5q^{-1}+13q^{-2}-3q^{-3}-6q^{-4}-9q^{-6}+6q^{-7}-3q^{-8}+18q^{-9}-2q^{-10}-6q^{-11}-q^{-12}-12q^{-13}+3q^{-14}-3q^{-15}+20q^{-16}-q^{-17}-5q^{-18}-11q^{-20}+2q^{-21}-4q^{-22}+18q^{-23}-2q^{-24}-5q^{-25}+q^{-26}-10q^{-27}+3q^{-28}-5q^{-29}+16q^{-30}-2q^{-31}-4q^{-32}+2q^{-33}-9q^{-34}+3q^{-35}-7q^{-36}+14q^{-37}-q^{-39}+3q^{-40}-9q^{-41}+2q^{-42}-10q^{-43}+11q^{-44}+3q^{-45}+2q^{-46}+4q^{-47}-9q^{-48}-13q^{-50}+9q^{-51}+5q^{-52}+5q^{-53}+5q^{-54}-9q^{-55}-q^{-56}-15q^{-57}+6q^{-58}+6q^{-59}+7q^{-60}+7q^{-61}-7q^{-62}-q^{-63}-15q^{-64}+2q^{-65}+4q^{-66}+7q^{-67}+8q^{-68}-4q^{-69}+q^{-70}-14q^{-71}-q^{-72}+q^{-73}+5q^{-74}+7q^{-75}-q^{-76}+5q^{-77}-11q^{-78}-2q^{-79}-q^{-80}+2q^{-81}+4q^{-82}+7q^{-84}-8q^{-85}-q^{-86}-q^{-87}+q^{-89}+7q^{-91}-7q^{-92}+7q^{-98}-6q^{-99}-q^{-100}-q^{-101}+q^{-104}+7q^{-105}-4q^{-106}-q^{-107}-2q^{-108}-q^{-109}-q^{-110}+6q^{-112}-q^{-113}-q^{-115}-q^{-116}-2q^{-117}-q^{-118}+3q^{-119}+q^{-121}-q^{-124}-q^{-125}+q^{-126}$
7	Not Available

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[8, 1]]
Out[2]=	PD[X[1, 4, 2, 5], X[9, 12, 10, 13], X[3, 11, 4, 10], X[11, 3, 12, 2], X[5, 16, 6, 1], X[7, 14, 8, 15], X[13, 8, 14, 9], X[15, 6, 16, 7]]
In[3]:=	GaussCode[Knot[8, 1]]
Out[3]=	GaussCode[-1, 4, -3, 1, -5, 8, -6, 7, -2, 3, -4, 2, -7, 6, -8, 5]
In[4]:=	DTCode[Knot[8, 1]]
Out[4]=	DTCode[4, 10, 16, 14, 12, 2, 8, 6]
In[5]:=	br = BR[Knot[8, 1]]
Out[5]=	BR[5, {-1, -1, -2, 1, -2, -3, 2, 4, -3, 4}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{5, 10}
In[7]:=	BraidIndex[Knot[8, 1]]
Out[7]=	5
In[8]:=	Show[DrawMorseLink[Knot[8, 1]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[8, 1]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{Reversible, 1, 1, 2, {4, 5}, 1}
In[10]:=	alex = Alexander[Knot[8, 1]][t]
Out[10]=	3 7 - - - 3 t t
In[11]:=	Conway[Knot[8, 1]][z]
Out[11]=	2 1 - 3 z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[8, 1]}
In[13]:=	{KnotDet[Knot[8, 1]], KnotSignature[Knot[8, 1]]}
Out[13]=	{13, 0}
In[14]:=	Jones[Knot[8, 1]][q]
Out[14]=	-6 -5 -4 2 2 2 2 2 + q - q + q - -- + -- - - - q + q 3 2 q q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[8, 1], Knot[11, NonAlternating, 70]}
In[16]:=	A2Invariant[Knot[8, 1]][q]
Out[16]=	-20 -18 -12 -10 2 6 8 q + q - q - q + q + q + q
In[17]:=	HOMFLYPT[Knot[8, 1]][a, z]
Out[17]=	-2 4 6 2 2 2 4 2 a - a + a - z - a z - a z
In[18]:=	Kauffman[Knot[8, 1]][a, z]
Out[18]=	2 3 -2 4 6 3 5 z 4 2 6 2 z 3 -a - a - a - 3 a z - 3 a z + -- + 7 a z + 6 a z + -- - a z + 2 a a 3 3 5 3 4 2 4 4 4 6 4 5 5 a z + 7 a z + z - 2 a z - 8 a z - 5 a z + a z - 3 5 5 5 2 6 4 6 6 6 3 7 5 7 4 a z - 5 a z + a z + 2 a z + a z + a z + a z
In[19]:=	{Vassiliev[2][Knot[8, 1]], Vassiliev[3][Knot[8, 1]]}
Out[19]=	{-3, 3}
In[20]:=	Kh[Knot[8, 1]][q, t]
Out[20]=	1 1 1 1 1 1 1 1 - + 2 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 9 5 9 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 1 1 5 2 ---- + --- + q t + q t 3 q t q t
In[21]:=	ColouredJones[Knot[8, 1], 2][q]
Out[21]=	-18 -17 -16 2 -14 2 3 3 3 3 3 2 + q - q - q + --- - q - --- + --- - --- + -- - -- + -- + 15 13 12 10 9 7 6 q q q q q q q -5 3 2 -2 3 2 3 5 6 q - -- + -- + q - - - 2 q + 2 q - q + q 4 3 q q q

See/edit the Rolfsen_Splice_Template.