# 10 93

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 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 93's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 93 at Knotilus!

### Knot presentations

 Planar diagram presentation X6271 X16,6,17,5 X20,8,1,7 X18,13,19,14 X14,9,15,10 X10,3,11,4 X4,11,5,12 X12,17,13,18 X8,20,9,19 X2,16,3,15 Gauss code 1, -10, 6, -7, 2, -1, 3, -9, 5, -6, 7, -8, 4, -5, 10, -2, 8, -4, 9, -3 Dowker-Thistlethwaite code 6 10 16 20 14 4 18 2 12 8 Conway Notation [.3.20.2]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 11, width is 4,

Braid index is 4

[{3, 8}, {9, 7}, {8, 12}, {2, 6}, {10, 13}, {11, 9}, {4, 10}, {6, 11}, {5, 3}, {12, 4}, {1, 5}, {13, 2}, {7, 1}]

### Three dimensional invariants

 Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-8][-4] Hyperbolic Volume 13.0165 A-Polynomial See Data:10 93/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle 3}$ Rasmussen s-Invariant 2

### Polynomial invariants

 Alexander polynomial ${\displaystyle 2t^{3}-8t^{2}+15t-17+15t^{-1}-8t^{-2}+2t^{-3}}$ Conway polynomial ${\displaystyle 2z^{6}+4z^{4}+z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 67, -2 } Jones polynomial ${\displaystyle -q^{4}+3q^{3}-5q^{2}+8q-10+11q^{-1}-10q^{-2}+9q^{-3}-6q^{-4}+3q^{-5}-q^{-6}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle a^{2}z^{6}+z^{6}-a^{4}z^{4}+3a^{2}z^{4}-z^{4}a^{-2}+3z^{4}-2a^{4}z^{2}+3a^{2}z^{2}-2z^{2}a^{-2}+2z^{2}-a^{4}+2a^{2}}$ Kauffman polynomial (db, data sources) ${\displaystyle 2az^{9}+2z^{9}a^{-1}+6a^{2}z^{8}+3z^{8}a^{-2}+9z^{8}+9a^{3}z^{7}+5az^{7}-3z^{7}a^{-1}+z^{7}a^{-3}+9a^{4}z^{6}-9a^{2}z^{6}-13z^{6}a^{-2}-31z^{6}+6a^{5}z^{5}-17a^{3}z^{5}-29az^{5}-10z^{5}a^{-1}-4z^{5}a^{-3}+3a^{6}z^{4}-14a^{4}z^{4}-6a^{2}z^{4}+17z^{4}a^{-2}+28z^{4}+a^{7}z^{3}-4a^{5}z^{3}+7a^{3}z^{3}+25az^{3}+18z^{3}a^{-1}+5z^{3}a^{-3}+7a^{4}z^{2}+7a^{2}z^{2}-6z^{2}a^{-2}-6z^{2}+a^{5}z-a^{3}z-6az-6za^{-1}-2za^{-3}-a^{4}-2a^{2}}$ The A2 invariant ${\displaystyle -q^{18}+q^{16}-q^{14}-q^{12}+2q^{10}-q^{8}+3q^{6}+1-2q^{-2}+2q^{-4}+q^{-10}-q^{-12}}$ The G2 invariant ${\displaystyle q^{100}-2q^{98}+3q^{96}-4q^{94}+3q^{92}-2q^{90}-q^{88}+7q^{86}-11q^{84}+15q^{82}-18q^{80}+13q^{78}-6q^{76}-6q^{74}+23q^{72}-33q^{70}+39q^{68}-37q^{66}+22q^{64}-2q^{62}-26q^{60}+49q^{58}-63q^{56}+61q^{54}-43q^{52}+7q^{50}+37q^{48}-69q^{46}+80q^{44}-60q^{42}+11q^{40}+38q^{38}-70q^{36}+69q^{34}-26q^{32}-31q^{30}+85q^{28}-98q^{26}+64q^{24}+7q^{22}-85q^{20}+135q^{18}-133q^{16}+85q^{14}-5q^{12}-69q^{10}+123q^{8}-132q^{6}+97q^{4}-37q^{2}-33+80q^{-2}-93q^{-4}+70q^{-6}-15q^{-8}-39q^{-10}+75q^{-12}-78q^{-14}+39q^{-16}+27q^{-18}-87q^{-20}+113q^{-22}-92q^{-24}+32q^{-26}+43q^{-28}-95q^{-30}+111q^{-32}-86q^{-34}+36q^{-36}+15q^{-38}-54q^{-40}+62q^{-42}-47q^{-44}+24q^{-46}-q^{-48}-12q^{-50}+13q^{-52}-11q^{-54}+6q^{-56}-2q^{-58}+q^{-60}}$

### "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: (1, -1)
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 ${\displaystyle 4}$ ${\displaystyle -8}$ ${\displaystyle 8}$ ${\displaystyle -{\frac {34}{3}}}$ ${\displaystyle -{\frac {86}{3}}}$ ${\displaystyle -32}$ ${\displaystyle -{\frac {176}{3}}}$ ${\displaystyle {\frac {160}{3}}}$ ${\displaystyle -72}$ ${\displaystyle {\frac {32}{3}}}$ ${\displaystyle 32}$ ${\displaystyle -{\frac {136}{3}}}$ ${\displaystyle -{\frac {344}{3}}}$ ${\displaystyle {\frac {751}{30}}}$ ${\displaystyle {\frac {698}{15}}}$ ${\displaystyle -{\frac {3658}{45}}}$ ${\displaystyle -{\frac {655}{18}}}$ ${\displaystyle -{\frac {2129}{30}}}$

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 ${\displaystyle s=}$-2 is the signature of 10 93. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-1012345χ
9          1-1
7         2 2
5        31 -2
3       52  3
1      53   -2
-1     65    1
-3    56     1
-5   45      -1
-7  25       3
-9 14        -3
-11 2         2
-131          -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$