# 7 2

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 7 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 2 at Knotilus!

### Knot presentations

 Planar diagram presentation X1425 X3,10,4,11 X5,14,6,1 X7,12,8,13 X11,8,12,9 X13,6,14,7 X9,2,10,3 Gauss code -1, 7, -2, 1, -3, 6, -4, 5, -7, 2, -5, 4, -6, 3 Dowker-Thistlethwaite code 4 10 14 12 2 8 6 Conway Notation [52]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 9, width is 4,

Braid index is 4

[{9, 6}, {5, 7}, {6, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 8}, {7, 9}, {8, 1}]
 Knot 7_2. A graph, knot 7_2.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 1 Bridge index 2 Super bridge index ${\displaystyle \{3,4\}}$ Nakanishi index 1 Maximal Thurston-Bennequin number $\displaystyle \text{\Failed}$ Hyperbolic Volume 3.33174 A-Polynomial See Data:7 2/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle {\textrm {ConcordanceGenus}}({\textrm {Knot}}(7,2))}$ Rasmussen s-Invariant -2

### Polynomial invariants

 Alexander polynomial ${\displaystyle 3t+3t^{-1}-5}$ Conway polynomial ${\displaystyle 3z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 11, -2 } Jones polynomial ${\displaystyle -q^{-8}+q^{-7}-q^{-6}+2q^{-5}-2q^{-4}+2q^{-3}-q^{-2}+q^{-1}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -a^{8}+a^{6}z^{2}+a^{6}+a^{4}z^{2}+a^{2}z^{2}+a^{2}}$ Kauffman polynomial (db, data sources) ${\displaystyle a^{9}z^{5}-4a^{9}z^{3}+3a^{9}z+a^{8}z^{6}-4a^{8}z^{4}+4a^{8}z^{2}-a^{8}+2a^{7}z^{5}-6a^{7}z^{3}+3a^{7}z+a^{6}z^{6}-3a^{6}z^{4}+3a^{6}z^{2}-a^{6}+a^{5}z^{5}-a^{5}z^{3}+a^{4}z^{4}+a^{3}z^{3}+a^{2}z^{2}-a^{2}}$ The A2 invariant ${\displaystyle -q^{26}-q^{24}+q^{18}+q^{16}+q^{8}+q^{6}+q^{2}}$ The G2 invariant ${\displaystyle q^{128}+q^{124}-q^{122}-q^{116}+2q^{114}-2q^{112}-q^{110}-q^{106}-2q^{102}-2q^{100}-q^{92}-q^{90}+2q^{88}+q^{78}+3q^{74}+q^{70}+q^{68}-q^{66}+2q^{64}-q^{60}+q^{54}-q^{50}-q^{40}+q^{38}+q^{36}+q^{34}+q^{28}+q^{24}+q^{20}+q^{14}+q^{10}}$

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {K11n88,}

### Vassiliev invariants

 V2 and V3: (3, -6)
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 ${\displaystyle 12}$ ${\displaystyle -48}$ ${\displaystyle 72}$ ${\displaystyle 222}$ ${\displaystyle 34}$ ${\displaystyle -576}$ ${\displaystyle -1152}$ ${\displaystyle -192}$ ${\displaystyle -176}$ ${\displaystyle 288}$ ${\displaystyle 1152}$ ${\displaystyle 2664}$ ${\displaystyle 408}$ ${\displaystyle {\frac {60431}{10}}}$ ${\displaystyle -{\frac {826}{15}}}$ ${\displaystyle {\frac {38942}{15}}}$ ${\displaystyle {\frac {497}{6}}}$ ${\displaystyle {\frac {3471}{10}}}$

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 7 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-7-6-5-4-3-2-10χ
-1       11
-3      110
-5     1  1
-7    11  0
-9   11   0
-11   1    1
-13 11     0
-15        0
-171       -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$