# 10 75

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 75's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 75 at Knotilus!
Decorative representation by Petr Vodicka.
Symmetrical decorative form made from 45-degree lines and circular arcs.
Decorative heart knot.

### Knot presentations

 Planar diagram presentation X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,16,14,17 X7,15,8,14 X15,7,16,6 X17,20,18,1 X9,19,10,18 X19,9,20,8 Gauss code -1, 4, -3, 1, -2, 7, -6, 10, -9, 3, -4, 2, -5, 6, -7, 5, -8, 9, -10, 8 Dowker-Thistlethwaite code 4 10 12 14 18 2 16 6 20 8 Conway Notation [21,21,21+]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 12, width is 5,

Braid index is 5

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

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [-5][-7] Hyperbolic Volume 13.4307 A-Polynomial See Data:10 75/A-polynomial

### Four dimensional invariants

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

### Polynomial invariants

 Alexander polynomial ${\displaystyle -t^{3}+7t^{2}-19t+27-19t^{-1}+7t^{-2}-t^{-3}}$ Conway polynomial ${\displaystyle -z^{6}+z^{4}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{3,t+1\}}$ Determinant and Signature { 81, 0 } Jones polynomial ${\displaystyle q^{6}-3q^{5}+6q^{4}-10q^{3}+12q^{2}-13q+14-10q^{-1}+7q^{-2}-4q^{-3}+q^{-4}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{6}+a^{2}z^{4}+3z^{4}a^{-2}-3z^{4}+a^{2}z^{2}+6z^{2}a^{-2}-3z^{2}a^{-4}-4z^{2}+3a^{-2}-3a^{-4}+a^{-6}}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{9}a^{-1}+z^{9}a^{-3}+7z^{8}a^{-2}+3z^{8}a^{-4}+4z^{8}+7az^{7}+13z^{7}a^{-1}+9z^{7}a^{-3}+3z^{7}a^{-5}+7a^{2}z^{6}-4z^{6}a^{-2}-3z^{6}a^{-4}+z^{6}a^{-6}+7z^{6}+4a^{3}z^{5}-5az^{5}-29z^{5}a^{-1}-29z^{5}a^{-3}-9z^{5}a^{-5}+a^{4}z^{4}-8a^{2}z^{4}-21z^{4}a^{-2}-9z^{4}a^{-4}-3z^{4}a^{-6}-24z^{4}-3a^{3}z^{3}-az^{3}+17z^{3}a^{-1}+24z^{3}a^{-3}+9z^{3}a^{-5}+4a^{2}z^{2}+20z^{2}a^{-2}+12z^{2}a^{-4}+3z^{2}a^{-6}+15z^{2}-az-5za^{-1}-7za^{-3}-3za^{-5}-3a^{-2}-3a^{-4}-a^{-6}}$ The A2 invariant ${\displaystyle q^{12}-2q^{10}+q^{8}-3q^{4}+4q^{2}+3q^{-2}+q^{-4}-q^{-6}+2q^{-8}-3q^{-10}-2q^{-16}+q^{-18}+q^{-20}}$ The G2 invariant ${\displaystyle q^{66}-3q^{64}+6q^{62}-10q^{60}+9q^{58}-6q^{56}-2q^{54}+19q^{52}-32q^{50}+50q^{48}-55q^{46}+39q^{44}-9q^{42}-40q^{40}+87q^{38}-126q^{36}+134q^{34}-107q^{32}+37q^{30}+62q^{28}-144q^{26}+193q^{24}-183q^{22}+114q^{20}-13q^{18}-96q^{16}+154q^{14}-150q^{12}+86q^{10}+31q^{8}-114q^{6}+133q^{4}-79q^{2}-29+154q^{-2}-237q^{-4}+223q^{-6}-125q^{-8}-26q^{-10}+202q^{-12}-306q^{-14}+309q^{-16}-212q^{-18}+55q^{-20}+106q^{-22}-224q^{-24}+248q^{-26}-183q^{-28}+72q^{-30}+62q^{-32}-144q^{-34}+145q^{-36}-70q^{-38}-41q^{-40}+132q^{-42}-177q^{-44}+133q^{-46}-32q^{-48}-92q^{-50}+197q^{-52}-228q^{-54}+181q^{-56}-76q^{-58}-50q^{-60}+136q^{-62}-175q^{-64}+153q^{-66}-90q^{-68}+16q^{-70}+43q^{-72}-71q^{-74}+70q^{-76}-46q^{-78}+22q^{-80}-11q^{-84}+12q^{-86}-10q^{-88}+6q^{-90}-2q^{-92}+q^{-94}}$

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

Same Alexander/Conway Polynomial: {10_42,}

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

### Vassiliev invariants

 V2 and V3: (0, -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 ${\displaystyle 0}$ ${\displaystyle -8}$ ${\displaystyle 0}$ ${\displaystyle -32}$ ${\displaystyle -8}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {272}{3}}}$ ${\displaystyle -{\frac {32}{3}}}$ ${\displaystyle -40}$ ${\displaystyle 0}$ ${\displaystyle 32}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle -96}$ ${\displaystyle {\frac {584}{3}}}$ ${\displaystyle -{\frac {760}{3}}}$ ${\displaystyle -{\frac {64}{3}}}$ ${\displaystyle -48}$

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 75. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-4-3-2-10123456χ
13          11
11         2 -2
9        41 3
7       62  -4
5      64   2
3     76    -1
1    76     1
-1   48      4
-3  36       -3
-5 14        3
-7 3         -3
-91          1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{7}}$ ${\displaystyle {\mathbb {Z} }^{7}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$