# 9 32

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 9 32's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 32 at Knotilus!

### Knot presentations

 Planar diagram presentation X1425 X13,18,14,1 X3948 X9,3,10,2 X7,15,8,14 X15,11,16,10 X5,12,6,13 X11,17,12,16 X17,7,18,6 Gauss code -1, 4, -3, 1, -7, 9, -5, 3, -4, 6, -8, 7, -2, 5, -6, 8, -9, 2 Dowker-Thistlethwaite code 4 8 12 14 2 16 18 10 6 Conway Notation [.21.20]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 9, width is 4,

Braid index is 4

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

### Three dimensional invariants

 Symmetry type Chiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index ${\displaystyle \{4,6\}}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-2][-9] Hyperbolic Volume 13.0999 A-Polynomial See Data:9 32/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 ${\displaystyle t^{3}-6t^{2}+14t-17+14t^{-1}-6t^{-2}+t^{-3}}$ Conway polynomial ${\displaystyle z^{6}-z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 59, 2 } Jones polynomial ${\displaystyle q^{7}-3q^{6}+6q^{5}-9q^{4}+10q^{3}-10q^{2}+9q-6+4q^{-1}-q^{-2}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle z^{6}a^{-2}+3z^{4}a^{-2}-2z^{4}a^{-4}-z^{4}+3z^{2}a^{-2}-4z^{2}a^{-4}+z^{2}a^{-6}-z^{2}+a^{-2}-2a^{-4}+a^{-6}+1}$ Kauffman polynomial (db, data sources) ${\displaystyle 2z^{8}a^{-2}+2z^{8}a^{-4}+5z^{7}a^{-1}+10z^{7}a^{-3}+5z^{7}a^{-5}+6z^{6}a^{-2}+7z^{6}a^{-4}+5z^{6}a^{-6}+4z^{6}+az^{5}-9z^{5}a^{-1}-18z^{5}a^{-3}-5z^{5}a^{-5}+3z^{5}a^{-7}-19z^{4}a^{-2}-18z^{4}a^{-4}-6z^{4}a^{-6}+z^{4}a^{-8}-8z^{4}-az^{3}+3z^{3}a^{-1}+9z^{3}a^{-3}+2z^{3}a^{-5}-3z^{3}a^{-7}+10z^{2}a^{-2}+12z^{2}a^{-4}+4z^{2}a^{-6}-z^{2}a^{-8}+3z^{2}-za^{-1}-2za^{-3}+za^{-7}-a^{-2}-2a^{-4}-a^{-6}+1}$ The A2 invariant ${\displaystyle -q^{6}+2q^{4}+1+3q^{-2}-2q^{-4}+2q^{-6}-2q^{-8}-2q^{-14}+2q^{-16}-q^{-18}+q^{-22}}$ The G2 invariant ${\displaystyle q^{32}-3q^{30}+7q^{28}-13q^{26}+13q^{24}-9q^{22}-6q^{20}+30q^{18}-50q^{16}+66q^{14}-56q^{12}+17q^{10}+39q^{8}-93q^{6}+126q^{4}-112q^{2}+58+22q^{-2}-92q^{-4}+126q^{-6}-106q^{-8}+48q^{-10}+29q^{-12}-83q^{-14}+89q^{-16}-47q^{-18}-23q^{-20}+92q^{-22}-122q^{-24}+101q^{-26}-35q^{-28}-53q^{-30}+131q^{-32}-173q^{-34}+158q^{-36}-91q^{-38}-6q^{-40}+98q^{-42}-157q^{-44}+157q^{-46}-103q^{-48}+19q^{-50}+58q^{-52}-102q^{-54}+89q^{-56}-33q^{-58}-39q^{-60}+90q^{-62}-94q^{-64}+49q^{-66}+22q^{-68}-90q^{-70}+125q^{-72}-111q^{-74}+63q^{-76}+q^{-78}-59q^{-80}+88q^{-82}-86q^{-84}+64q^{-86}-26q^{-88}-5q^{-90}+25q^{-92}-33q^{-94}+30q^{-96}-21q^{-98}+12q^{-100}-2q^{-102}-4q^{-104}+5q^{-106}-6q^{-108}+4q^{-110}-2q^{-112}+q^{-114}}$

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

Same Alexander/Conway Polynomial: {K11n52, K11n124,}

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

### Vassiliev invariants

 V2 and V3: (-1, -2)
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 ${\displaystyle -4}$ ${\displaystyle -16}$ ${\displaystyle 8}$ ${\displaystyle -{\frac {62}{3}}}$ ${\displaystyle {\frac {14}{3}}}$ ${\displaystyle 64}$ ${\displaystyle {\frac {224}{3}}}$ ${\displaystyle {\frac {128}{3}}}$ ${\displaystyle 16}$ ${\displaystyle -{\frac {32}{3}}}$ ${\displaystyle 128}$ ${\displaystyle {\frac {248}{3}}}$ ${\displaystyle -{\frac {56}{3}}}$ ${\displaystyle {\frac {12689}{30}}}$ ${\displaystyle {\frac {74}{5}}}$ ${\displaystyle {\frac {9778}{45}}}$ ${\displaystyle {\frac {79}{18}}}$ ${\displaystyle {\frac {689}{30}}}$

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 9 32. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-3-2-10123456χ
15         11
13        2 -2
11       41 3
9      52  -3
7     54   1
5    55    0
3   45     -1
1  36      3
-1 13       -2
-3 3        3
-51         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$