# 9 42

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 9 42's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 42 at Knotilus!

9_42 is Alexander Stoimenow's favourite knot!

 Alsacian chair, alsacian museum, Strasbourg, France

### Knot presentations

 Planar diagram presentation X1425 X5,10,6,11 X3948 X9,3,10,2 X16,12,17,11 X14,7,15,8 X6,15,7,16 X18,14,1,13 X12,18,13,17 Gauss code -1, 4, -3, 1, -2, -7, 6, 3, -4, 2, 5, -9, 8, -6, 7, -5, 9, -8 Dowker-Thistlethwaite code 4 8 10 -14 2 -16 -18 -6 -12 Conway Notation [22,3,2-]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 9, width is 4,

Braid index is 4

[{11, 2}, {1, 9}, {10, 5}, {9, 11}, {8, 4}, {2, 7}, {6, 8}, {7, 10}, {5, 3}, {4, 1}, {3, 6}]
 Knot 9_42. A graph, knot 9_42. A part of a knot and a part of a graph.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number [-3][-5] Hyperbolic Volume 4.05686 A-Polynomial See Data:9 42/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 ${\displaystyle -t^{2}+2t-1+2t^{-1}-t^{-2}}$ Conway polynomial ${\displaystyle -z^{4}-2z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 7, 2 } Jones polynomial ${\displaystyle q^{3}-q^{2}+q-1+q^{-1}-q^{-2}+q^{-3}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{4}+a^{2}z^{2}+z^{2}a^{-2}-4z^{2}+2a^{2}+2a^{-2}-3}$ Kauffman polynomial (db, data sources) ${\displaystyle az^{7}+z^{7}a^{-1}+a^{2}z^{6}+z^{6}a^{-2}+2z^{6}-5az^{5}-5z^{5}a^{-1}-5a^{2}z^{4}-5z^{4}a^{-2}-10z^{4}+6az^{3}+6z^{3}a^{-1}+6a^{2}z^{2}+6z^{2}a^{-2}+12z^{2}-2az-2za^{-1}-2a^{2}-2a^{-2}-3}$ The A2 invariant ${\displaystyle q^{10}+q^{8}+q^{6}-q^{2}-1-q^{-2}+q^{-6}+q^{-8}+q^{-10}}$ The G2 invariant ${\displaystyle q^{46}+q^{42}+2q^{32}+q^{26}+q^{24}+q^{22}+q^{20}-q^{18}+q^{16}+q^{14}-q^{12}+q^{10}-q^{8}-q^{4}-2q^{2}-1-q^{-2}-q^{-4}-2q^{-6}-q^{-8}-q^{-10}+q^{-12}-q^{-14}-q^{-16}+q^{-20}+q^{-22}+q^{-24}+q^{-26}+3q^{-30}+q^{-34}+q^{-36}+q^{-40}+q^{-46}-q^{-50}-q^{-54}+q^{-56}-q^{-60}+q^{-62}}$

### "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 ${\displaystyle -8}$ ${\displaystyle 0}$ ${\displaystyle 32}$ ${\displaystyle {\frac {164}{3}}}$ ${\displaystyle {\frac {76}{3}}}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {256}{3}}}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {1312}{3}}}$ ${\displaystyle -{\frac {608}{3}}}$ ${\displaystyle -{\frac {6271}{15}}}$ ${\displaystyle {\frac {1484}{15}}}$ ${\displaystyle -{\frac {19564}{45}}}$ ${\displaystyle {\frac {607}{9}}}$ ${\displaystyle -{\frac {1471}{15}}}$

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=}$2 is the signature of 9 42. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-4-3-2-1012χ
7      11
5       0
3    11 0
1   11  0
-1   11  0
-3 11    0
-5       0
-71      1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$