# 8 1

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 8 1's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 1 at Knotilus!

### 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 A Morse Link Presentation An Arc Presentation

Length is 10, width is 5,

Braid index is 5

[{10, 7}, {6, 8}, {7, 5}, {4, 6}, {5, 3}, {2, 4}, {3, 1}, {9, 2}, {8, 10}, {1, 9}]
 Knot 8_1. A graph, knot 8_1.

### Three dimensional invariants

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

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle 1}$ Rasmussen s-Invariant 0

### Polynomial invariants

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

### "Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {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 ${\displaystyle -12}$ ${\displaystyle 24}$ ${\displaystyle 72}$ ${\displaystyle 66}$ ${\displaystyle 38}$ ${\displaystyle -288}$ ${\displaystyle -432}$ ${\displaystyle -96}$ ${\displaystyle -104}$ ${\displaystyle -288}$ ${\displaystyle 288}$ ${\displaystyle -792}$ ${\displaystyle -456}$ ${\displaystyle {\frac {1009}{10}}}$ ${\displaystyle {\frac {2626}{15}}}$ ${\displaystyle -{\frac {6182}{15}}}$ ${\displaystyle {\frac {943}{6}}}$ ${\displaystyle -{\frac {1551}{10}}}$

V2,1 through V6,9 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 ${\displaystyle t^{r}q^{j}}$ are shown, along with their alternating sums ${\displaystyle \chi }$ (fixed ${\displaystyle j}$, alternation over ${\displaystyle r}$). The squares with yellow highlighting are those on the "critical diagonals", where ${\displaystyle j-2r=s+1}$ or ${\displaystyle j-2r=s-1}$, where ${\displaystyle s=}$0 is the signature of 8 1. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-6-5-4-3-2-1012χ
5        11
3         0
1      21 1
-1     11  0
-3    11   0
-5   11    0
-7   1     -1
-9 11      0
-11         0
-131        1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$