# 10 17

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 17's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 17 at Knotilus!

### Knot presentations

 Planar diagram presentation X6271 X12,4,13,3 X20,15,1,16 X16,7,17,8 X18,9,19,10 X8,17,9,18 X10,19,11,20 X14,6,15,5 X2,12,3,11 X4,14,5,13 Gauss code 1, -9, 2, -10, 8, -1, 4, -6, 5, -7, 9, -2, 10, -8, 3, -4, 6, -5, 7, -3 Dowker-Thistlethwaite code 6 12 14 16 18 2 4 20 8 10 Conway Notation [4114]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 10, width is 3,

Braid index is 3

[{4, 12}, {3, 6}, {5, 7}, {6, 8}, {7, 9}, {8, 11}, {9, 4}, {2, 5}, {12, 10}, {1, 3}, {11, 2}, {10, 1}]

### Three dimensional invariants

 Symmetry type Fully amphicheiral Unknotting number 1 3-genus 4 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 8.53676 A-Polynomial See Data:10 17/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial ${\displaystyle t^{4}-3t^{3}+5t^{2}-7t+9-7t^{-1}+5t^{-2}-3t^{-3}+t^{-4}}$ Conway polynomial ${\displaystyle z^{8}+5z^{6}+7z^{4}+2z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 41, 0 } Jones polynomial ${\displaystyle -q^{5}+2q^{4}-3q^{3}+5q^{2}-6q+7-6q^{-1}+5q^{-2}-3q^{-3}+2q^{-4}-q^{-5}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{8}-a^{2}z^{6}-z^{6}a^{-2}+7z^{6}-5a^{2}z^{4}-5z^{4}a^{-2}+17z^{4}-7a^{2}z^{2}-7z^{2}a^{-2}+16z^{2}-2a^{2}-2a^{-2}+5}$ Kauffman polynomial (db, data sources) ${\displaystyle az^{9}+z^{9}a^{-1}+2a^{2}z^{8}+2z^{8}a^{-2}+4z^{8}+2a^{3}z^{7}-2az^{7}-2z^{7}a^{-1}+2z^{7}a^{-3}+2a^{4}z^{6}-7a^{2}z^{6}-7z^{6}a^{-2}+2z^{6}a^{-4}-18z^{6}+a^{5}z^{5}-5a^{3}z^{5}-5z^{5}a^{-3}+z^{5}a^{-5}-6a^{4}z^{4}+11a^{2}z^{4}+11z^{4}a^{-2}-6z^{4}a^{-4}+34z^{4}-3a^{5}z^{3}+2a^{3}z^{3}+6az^{3}+6z^{3}a^{-1}+2z^{3}a^{-3}-3z^{3}a^{-5}+3a^{4}z^{2}-8a^{2}z^{2}-8z^{2}a^{-2}+3z^{2}a^{-4}-22z^{2}+a^{5}z-3az-3za^{-1}+za^{-5}+2a^{2}+2a^{-2}+5}$ The A2 invariant ${\displaystyle -q^{14}-q^{10}+q^{8}+q^{6}+2q^{2}-1+2q^{-2}+q^{-6}+q^{-8}-q^{-10}-q^{-14}}$ The G2 invariant ${\displaystyle q^{80}-q^{78}+2q^{76}-3q^{74}+2q^{72}-q^{70}-2q^{68}+6q^{66}-7q^{64}+8q^{62}-7q^{60}+2q^{58}+2q^{56}-8q^{54}+11q^{52}-15q^{50}+12q^{48}-7q^{46}-2q^{44}+9q^{42}-16q^{40}+19q^{38}-14q^{36}+4q^{34}+5q^{32}-14q^{30}+15q^{28}-7q^{26}-q^{24}+11q^{22}-12q^{20}+9q^{18}+2q^{16}-12q^{14}+21q^{12}-20q^{10}+14q^{8}-q^{6}-12q^{4}+23q^{2}-25+23q^{-2}-12q^{-4}-q^{-6}+14q^{-8}-20q^{-10}+21q^{-12}-12q^{-14}+2q^{-16}+9q^{-18}-12q^{-20}+11q^{-22}-q^{-24}-7q^{-26}+15q^{-28}-14q^{-30}+5q^{-32}+4q^{-34}-14q^{-36}+19q^{-38}-16q^{-40}+9q^{-42}-2q^{-44}-7q^{-46}+12q^{-48}-15q^{-50}+11q^{-52}-8q^{-54}+2q^{-56}+2q^{-58}-7q^{-60}+8q^{-62}-7q^{-64}+6q^{-66}-2q^{-68}-q^{-70}+2q^{-72}-3q^{-74}+2q^{-76}-q^{-78}+q^{-80}}$

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

Same Alexander/Conway Polynomial: {}

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

### Vassiliev invariants

 V2 and V3: (2, 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 ${\displaystyle 8}$ ${\displaystyle 0}$ ${\displaystyle 32}$ ${\displaystyle -{\frac {20}{3}}}$ ${\displaystyle -{\frac {124}{3}}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle {\frac {256}{3}}}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {160}{3}}}$ ${\displaystyle -{\frac {992}{3}}}$ ${\displaystyle -{\frac {2009}{15}}}$ ${\displaystyle {\frac {3476}{15}}}$ ${\displaystyle -{\frac {25556}{45}}}$ ${\displaystyle {\frac {89}{9}}}$ ${\displaystyle -{\frac {2489}{15}}}$

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 10 17. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-1012345χ
11          1-1
9         1 1
7        21 -1
5       31  2
3      32   -1
1     43    1
-1    34     1
-3   23      -1
-5  13       2
-7 12        -1
-9 1         1
-111          -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$