# 10 33

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 10 33's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 10 33 at Knotilus!

### Knot presentations

 Planar diagram presentation X6271 X14,6,15,5 X20,15,1,16 X16,7,17,8 X8,19,9,20 X18,9,19,10 X10,17,11,18 X2,14,3,13 X12,4,13,3 X4,12,5,11 Gauss code 1, -8, 9, -10, 2, -1, 4, -5, 6, -7, 10, -9, 8, -2, 3, -4, 7, -6, 5, -3 Dowker-Thistlethwaite code 6 12 14 16 18 4 2 20 10 8 Conway Notation [311113]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 12, width is 5,

Braid index is 5

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

### Three dimensional invariants

 Symmetry type Fully amphicheiral Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 11.5357 A-Polynomial See Data:10 33/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle 2}$ Rasmussen s-Invariant 0

### Polynomial invariants

 Alexander polynomial ${\displaystyle 4t^{2}-16t+25-16t^{-1}+4t^{-2}}$ Conway polynomial ${\displaystyle 4z^{4}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 65, 0 } Jones polynomial ${\displaystyle -q^{5}+3q^{4}-5q^{3}+8q^{2}-10q+11-10q^{-1}+8q^{-2}-5q^{-3}+3q^{-4}-q^{-5}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{2}a^{4}+z^{4}a^{2}+2z^{4}+2z^{2}+1+z^{4}a^{-2}-z^{2}a^{-4}}$ Kauffman polynomial (db, data sources) ${\displaystyle az^{9}+z^{9}a^{-1}+3a^{2}z^{8}+3z^{8}a^{-2}+6z^{8}+4a^{3}z^{7}+5az^{7}+5z^{7}a^{-1}+4z^{7}a^{-3}+3a^{4}z^{6}-4a^{2}z^{6}-4z^{6}a^{-2}+3z^{6}a^{-4}-14z^{6}+a^{5}z^{5}-9a^{3}z^{5}-16az^{5}-16z^{5}a^{-1}-9z^{5}a^{-3}+z^{5}a^{-5}-7a^{4}z^{4}+a^{2}z^{4}+z^{4}a^{-2}-7z^{4}a^{-4}+16z^{4}-2a^{5}z^{3}+6a^{3}z^{3}+18az^{3}+18z^{3}a^{-1}+6z^{3}a^{-3}-2z^{3}a^{-5}+3a^{4}z^{2}+3z^{2}a^{-4}-6z^{2}-2a^{3}z-6az-6za^{-1}-2za^{-3}+1}$ The A2 invariant ${\displaystyle -q^{16}+q^{14}+q^{12}-2q^{10}+2q^{8}-q^{4}+2q^{2}-1+2q^{-2}-q^{-4}+2q^{-8}-2q^{-10}+q^{-12}+q^{-14}-q^{-16}}$ The G2 invariant ${\displaystyle q^{80}-2q^{78}+4q^{76}-7q^{74}+6q^{72}-5q^{70}-2q^{68}+14q^{66}-23q^{64}+32q^{62}-32q^{60}+19q^{58}+3q^{56}-33q^{54}+60q^{52}-73q^{50}+67q^{48}-37q^{46}-9q^{44}+57q^{42}-88q^{40}+96q^{38}-70q^{36}+20q^{34}+31q^{32}-69q^{30}+73q^{28}-46q^{26}+45q^{22}-65q^{20}+47q^{18}-3q^{16}-55q^{14}+102q^{12}-111q^{10}+80q^{8}-14q^{6}-61q^{4}+124q^{2}-145+124q^{-2}-61q^{-4}-14q^{-6}+80q^{-8}-111q^{-10}+102q^{-12}-55q^{-14}-3q^{-16}+47q^{-18}-65q^{-20}+45q^{-22}-46q^{-26}+73q^{-28}-69q^{-30}+31q^{-32}+20q^{-34}-70q^{-36}+96q^{-38}-88q^{-40}+57q^{-42}-9q^{-44}-37q^{-46}+67q^{-48}-73q^{-50}+60q^{-52}-33q^{-54}+3q^{-56}+19q^{-58}-32q^{-60}+32q^{-62}-23q^{-64}+14q^{-66}-2q^{-68}-5q^{-70}+6q^{-72}-7q^{-74}+4q^{-76}-2q^{-78}+q^{-80}}$

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

Same Alexander/Conway Polynomial: {K11a333,}

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

### Vassiliev invariants

 V2 and V3: (0, 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 ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle -32}$ ${\displaystyle -32}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle -16}$ ${\displaystyle -{\frac {64}{3}}}$ ${\displaystyle {\frac {128}{3}}}$ ${\displaystyle -{\frac {112}{3}}}$ ${\displaystyle -16}$

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 33. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-1012345χ
11          1-1
9         2 2
7        31 -2
5       52  3
3      53   -2
1     65    1
-1    56     1
-3   35      -2
-5  25       3
-7 13        -2
-9 2         2
-111          -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{6}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$