# 7 5

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 7 5's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 7 5 at Knotilus!

### Knot presentations

 Planar diagram presentation X1425 X3,10,4,11 X5,12,6,13 X7,14,8,1 X13,6,14,7 X11,8,12,9 X9,2,10,3 Gauss code -1, 7, -2, 1, -3, 5, -4, 6, -7, 2, -6, 3, -5, 4 Dowker-Thistlethwaite code 4 10 12 14 2 8 6 Conway Notation [322]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 8, width is 3,

Braid index is 3

[{9, 2}, {1, 7}, {6, 8}, {7, 9}, {8, 3}, {2, 4}, {3, 5}, {4, 6}, {5, 1}]
 Knot 7_5. A graph, knot 7_5.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number $\displaystyle \text{\Failed}$ Hyperbolic Volume 6.44354 A-Polynomial See Data:7 5/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 2}$ Topological 4 genus ${\displaystyle 2}$ Concordance genus ${\displaystyle {\textrm {ConcordanceGenus}}({\textrm {Knot}}(7,5))}$ Rasmussen s-Invariant -4

### Polynomial invariants

 Alexander polynomial ${\displaystyle 2t^{2}-4t+5-4t^{-1}+2t^{-2}}$ Conway polynomial ${\displaystyle 2z^{4}+4z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 17, -4 } Jones polynomial ${\displaystyle -q^{-9}+2q^{-8}-3q^{-7}+3q^{-6}-3q^{-5}+3q^{-4}-q^{-3}+q^{-2}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle a^{8}\left(-z^{2}\right)-a^{8}+a^{6}z^{4}+2a^{6}z^{2}+a^{4}z^{4}+3a^{4}z^{2}+2a^{4}}$ Kauffman polynomial (db, data sources) ${\displaystyle a^{11}z^{3}-a^{11}z+2a^{10}z^{4}-2a^{10}z^{2}+2a^{9}z^{5}-2a^{9}z^{3}+a^{9}z+a^{8}z^{6}+a^{8}z^{2}-a^{8}+3a^{7}z^{5}-4a^{7}z^{3}+a^{7}z+a^{6}z^{6}-a^{6}z^{4}+a^{5}z^{5}-a^{5}z^{3}-a^{5}z+a^{4}z^{4}-3a^{4}z^{2}+2a^{4}}$ The A2 invariant ${\displaystyle -q^{28}-q^{22}-q^{18}+q^{16}+q^{14}+q^{12}+2q^{10}+q^{6}}$ The G2 invariant ${\displaystyle q^{148}-q^{146}+2q^{144}-2q^{142}+q^{138}-2q^{136}+5q^{134}-5q^{132}+4q^{130}-2q^{128}-3q^{126}+4q^{124}-6q^{122}+5q^{120}-3q^{118}-q^{116}+3q^{114}-3q^{112}+q^{110}+2q^{108}-5q^{106}+4q^{104}-3q^{102}-2q^{100}+5q^{98}-7q^{96}+8q^{94}-7q^{92}+2q^{90}+2q^{88}-6q^{86}+6q^{84}-7q^{82}+4q^{80}-2q^{76}+3q^{74}-3q^{72}+2q^{70}+3q^{68}-5q^{66}+3q^{64}-2q^{60}+7q^{58}-5q^{56}+5q^{54}-q^{52}+4q^{48}-4q^{46}+5q^{44}-q^{42}+q^{40}+q^{38}-q^{36}+2q^{34}+q^{30}}$

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

Same Alexander/Conway Polynomial: {10_130,}

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

### Vassiliev invariants

 V2 and V3: (4, -8)
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 ${\displaystyle 16}$ ${\displaystyle -64}$ ${\displaystyle 128}$ ${\displaystyle {\frac {968}{3}}}$ ${\displaystyle {\frac {136}{3}}}$ ${\displaystyle -1024}$ ${\displaystyle -{\frac {5440}{3}}}$ ${\displaystyle -{\frac {928}{3}}}$ ${\displaystyle -224}$ ${\displaystyle {\frac {2048}{3}}}$ ${\displaystyle 2048}$ ${\displaystyle {\frac {15488}{3}}}$ ${\displaystyle {\frac {2176}{3}}}$ ${\displaystyle {\frac {156422}{15}}}$ ${\displaystyle {\frac {5912}{15}}}$ ${\displaystyle {\frac {170888}{45}}}$ ${\displaystyle {\frac {730}{9}}}$ ${\displaystyle {\frac {7142}{15}}}$

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 ${\displaystyle s=}$-4 is the signature of 7 5. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-7-6-5-4-3-2-10χ
-3       11
-5      110
-7     2  2
-9    11  0
-11   22   0
-13  11    0
-15 12     -1
-17 1      1
-191       -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-5}$ ${\displaystyle i=-3}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$