# 7 7

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

 Ornamental knot Mongolian ornament ; sum of two 7.7 Depiction with three loops Sum of 4.1 and 7.7

### Knot presentations

 Planar diagram presentation X1425 X5,10,6,11 X3948 X9,3,10,2 X11,14,12,1 X7,13,8,12 X13,7,14,6 Gauss code -1, 4, -3, 1, -2, 7, -6, 3, -4, 2, -5, 6, -7, 5 Dowker-Thistlethwaite code 4 8 10 12 2 14 6 Conway Notation [21112]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 7, width is 4,

Braid index is 4

[{9, 3}, {2, 7}, {8, 4}, {3, 5}, {7, 9}, {4, 1}, {6, 2}, {5, 8}, {1, 6}]
 Knot 7_7. A graph, knot 7_7.

### Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index 4 Nakanishi index 1 Maximal Thurston-Bennequin number $\displaystyle \text{\Failed}$ Hyperbolic Volume 7.64338 A-Polynomial See Data:7 7/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 ${\displaystyle t^{2}+t^{-2}-5t-5t^{-1}+9}$ Conway polynomial ${\displaystyle z^{4}-z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 21, 0 } Jones polynomial ${\displaystyle q^{4}-2q^{3}-q^{-3}+3q^{2}+3q^{-2}-4q-3q^{-1}+4}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle a^{-4}-a^{2}z^{2}-2z^{2}a^{-2}-2a^{-2}+z^{4}+2z^{2}+2}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{4}a^{-4}-2z^{2}a^{-4}+a^{-4}+2z^{5}a^{-3}+a^{3}z^{3}-4z^{3}a^{-3}+2za^{-3}+z^{6}a^{-2}+3a^{2}z^{4}+2z^{4}a^{-2}-3a^{2}z^{2}-6z^{2}a^{-2}+2a^{-2}+3az^{5}+5z^{5}a^{-1}-3az^{3}-8z^{3}a^{-1}+az+3za^{-1}+z^{6}+4z^{4}-7z^{2}+2}$ The A2 invariant ${\displaystyle -q^{10}+q^{8}+q^{6}+2q^{2}+q^{-2}-q^{-4}-q^{-6}-q^{-10}+q^{-12}+q^{-14}}$ The G2 invariant ${\displaystyle q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{42}-4q^{40}+9q^{38}-9q^{36}+9q^{34}-3q^{32}-4q^{30}+9q^{28}-10q^{26}+9q^{24}-5q^{22}-q^{20}+5q^{18}-4q^{16}+4q^{14}+2q^{12}-7q^{10}+10q^{8}-5q^{6}-2q^{4}+8q^{2}-12+17q^{-2}-11q^{-4}+5q^{-6}+3q^{-8}-9q^{-10}+15q^{-12}-14q^{-14}+6q^{-16}-q^{-18}-4q^{-20}+6q^{-22}-6q^{-24}+q^{-26}+3q^{-28}-7q^{-30}+5q^{-32}-4q^{-34}-5q^{-36}+10q^{-38}-11q^{-40}+9q^{-42}-4q^{-44}-q^{-46}+7q^{-48}-8q^{-50}+9q^{-52}-4q^{-54}+q^{-56}+q^{-58}-3q^{-60}+3q^{-62}-q^{-64}+q^{-66}}$

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

Same Alexander/Conway Polynomial: {K11n28,}

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

### Vassiliev invariants

 V2 and V3: (-1, -1)
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 ${\displaystyle -4}$ ${\displaystyle -8}$ ${\displaystyle 8}$ ${\displaystyle -{\frac {14}{3}}}$ ${\displaystyle -{\frac {10}{3}}}$ ${\displaystyle 32}$ ${\displaystyle {\frac {112}{3}}}$ ${\displaystyle {\frac {64}{3}}}$ ${\displaystyle -8}$ ${\displaystyle -{\frac {32}{3}}}$ ${\displaystyle 32}$ ${\displaystyle {\frac {56}{3}}}$ ${\displaystyle {\frac {40}{3}}}$ ${\displaystyle {\frac {2849}{30}}}$ ${\displaystyle -{\frac {218}{15}}}$ ${\displaystyle {\frac {3418}{45}}}$ ${\displaystyle -{\frac {161}{18}}}$ ${\displaystyle {\frac {449}{30}}}$

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=}$0 is the signature of 7 7. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-3-2-101234χ
9       11
7      1 -1
5     21 1
3    21  -1
1   22   0
-1  23    1
-3 11     0
-5 2      2
-71       -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-1}$ ${\displaystyle i=1}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$