# 8 17

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 8 17's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 8 17 at Knotilus!
 A knot in Brian Sanderson's Garden [1]

### Knot presentations

 Planar diagram presentation X6271 X14,8,15,7 X8394 X2,13,3,14 X12,5,13,6 X4,9,5,10 X16,12,1,11 X10,16,11,15 Gauss code 1, -4, 3, -6, 5, -1, 2, -3, 6, -8, 7, -5, 4, -2, 8, -7 Dowker-Thistlethwaite code 6 8 12 14 4 16 2 10 Conway Notation [.2.2]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 8, width is 3,

Braid index is 3

[{7, 10}, {9, 2}, {10, 4}, {3, 5}, {4, 8}, {6, 9}, {5, 1}, {2, 7}, {1, 6}, {8, 3}]
 Knot 8_17. A graph, knot 8_17.

### Three dimensional invariants

 Symmetry type Negative amphicheiral Unknotting number 1 3-genus 3 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-5][-5] Hyperbolic Volume 10.9859 A-Polynomial See Data:8 17/A-polynomial

[edit Notes for 8 17's three dimensional invariants] 8_17 is the first negatively amphicheiral knot in the Rolfsen Table. Namely, it is equal to the inverse of its mirror, yet it is different from both its inverse and its mirror.

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle 3}$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial ${\displaystyle -t^{3}+4t^{2}-8t+11-8t^{-1}+4t^{-2}-t^{-3}}$ Conway polynomial ${\displaystyle -z^{6}-2z^{4}-z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 37, 0 } Jones polynomial ${\displaystyle q^{4}-3q^{3}+5q^{2}-6q+7-6q^{-1}+5q^{-2}-3q^{-3}+q^{-4}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{6}+a^{2}z^{4}+z^{4}a^{-2}-4z^{4}+2a^{2}z^{2}+2z^{2}a^{-2}-5z^{2}+a^{2}+a^{-2}-1}$ Kauffman polynomial (db, data sources) ${\displaystyle 2az^{7}+2z^{7}a^{-1}+4a^{2}z^{6}+4z^{6}a^{-2}+8z^{6}+3a^{3}z^{5}+2az^{5}+2z^{5}a^{-1}+3z^{5}a^{-3}+a^{4}z^{4}-6a^{2}z^{4}-6z^{4}a^{-2}+z^{4}a^{-4}-14z^{4}-4a^{3}z^{3}-6az^{3}-6z^{3}a^{-1}-4z^{3}a^{-3}-a^{4}z^{2}+3a^{2}z^{2}+3z^{2}a^{-2}-z^{2}a^{-4}+8z^{2}+a^{3}z+2az+2za^{-1}+za^{-3}-a^{2}-a^{-2}-1}$ The A2 invariant ${\displaystyle q^{12}-q^{10}+q^{8}-q^{4}+2q^{2}-1+2q^{-2}-q^{-4}+q^{-8}-q^{-10}+q^{-12}}$ The G2 invariant ${\displaystyle q^{66}-2q^{64}+4q^{62}-6q^{60}+4q^{58}-2q^{56}-4q^{54}+14q^{52}-21q^{50}+26q^{48}-20q^{46}+3q^{44}+17q^{42}-36q^{40}+47q^{38}-38q^{36}+17q^{34}+10q^{32}-32q^{30}+41q^{28}-29q^{26}+6q^{24}+18q^{22}-31q^{20}+23q^{18}-2q^{16}-24q^{14}+44q^{12}-47q^{10}+33q^{8}-5q^{6}-27q^{4}+53q^{2}-63+53q^{-2}-27q^{-4}-5q^{-6}+33q^{-8}-47q^{-10}+44q^{-12}-24q^{-14}-2q^{-16}+23q^{-18}-31q^{-20}+18q^{-22}+6q^{-24}-29q^{-26}+41q^{-28}-32q^{-30}+10q^{-32}+17q^{-34}-38q^{-36}+47q^{-38}-36q^{-40}+17q^{-42}+3q^{-44}-20q^{-46}+26q^{-48}-21q^{-50}+14q^{-52}-4q^{-54}-2q^{-56}+4q^{-58}-6q^{-60}+4q^{-62}-2q^{-64}+q^{-66}}$

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

Same Alexander/Conway Polynomial: {K11n53,}

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

### Vassiliev invariants

 V2 and V3: (-1, 0)
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 ${\displaystyle -4}$ ${\displaystyle 0}$ ${\displaystyle 8}$ ${\displaystyle {\frac {82}{3}}}$ ${\displaystyle {\frac {62}{3}}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {32}{3}}}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {328}{3}}}$ ${\displaystyle -{\frac {248}{3}}}$ ${\displaystyle -{\frac {2911}{30}}}$ ${\displaystyle {\frac {1742}{15}}}$ ${\displaystyle -{\frac {11342}{45}}}$ ${\displaystyle {\frac {991}{18}}}$ ${\displaystyle -{\frac {1951}{30}}}$

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 17. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-4-3-2-101234χ
9        11
7       2 -2
5      31 2
3     32  -1
1    43   1
-1   34    1
-3  23     -1
-5 13      2
-7 2       -2
-91        1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$