# 5 2

 (KnotPlot image) See the full Rolfsen Knot Table. Visit 5 2's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 5 2 at Knotilus! 5_2 is also known as the 3-twist knot.

 3D depiction Simple square depiction Lissajous curve x=cos(2t+0.2), y=cos(3t+0.7), z=cos(7t); 2 crossings can be removed

### Knot presentations

 Planar diagram presentation X1425 X3849 X5,10,6,1 X9,6,10,7 X7283 Gauss code -1, 5, -2, 1, -3, 4, -5, 2, -4, 3 Dowker-Thistlethwaite code 4 8 10 2 6 Conway Notation [32]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 6, width is 3,

Braid index is 3

[{7, 4}, {3, 5}, {4, 2}, {1, 3}, {2, 6}, {5, 7}, {6, 1}]
 Knot 5_2. A graph, knot 5_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 [-8][1] Hyperbolic Volume 2.82812 A-Polynomial See Data:5 2/A-polynomial

### Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle {\textrm {ConcordanceGenus}}({\textrm {Knot}}(5,2))}$ Rasmussen s-Invariant -2

### Polynomial invariants

 Alexander polynomial ${\displaystyle 2t+2t^{-1}-3}$ Conway polynomial ${\displaystyle 2z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 7, -2 } Jones polynomial ${\displaystyle -q^{-6}+q^{-5}-q^{-4}+2q^{-3}-q^{-2}+q^{-1}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -a^{6}+a^{4}z^{2}+a^{4}+a^{2}z^{2}+a^{2}}$ Kauffman polynomial (db, data sources) ${\displaystyle a^{7}z^{3}-2a^{7}z+a^{6}z^{4}-2a^{6}z^{2}+a^{6}+2a^{5}z^{3}-2a^{5}z+a^{4}z^{4}-a^{4}z^{2}+a^{4}+a^{3}z^{3}+a^{2}z^{2}-a^{2}}$ The A2 invariant ${\displaystyle -q^{20}-q^{18}+q^{12}+q^{10}+q^{8}+q^{6}+q^{2}}$ The G2 invariant ${\displaystyle q^{100}+q^{96}-q^{94}-q^{92}+q^{90}-q^{88}-q^{84}-q^{82}-q^{78}-q^{76}-q^{74}-q^{72}-q^{68}-q^{66}+q^{64}+q^{60}+q^{56}+q^{54}+2q^{50}-q^{48}+2q^{46}+q^{44}+q^{40}+q^{34}+2q^{24}+q^{20}+q^{14}+q^{10}}$

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

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {K11n57,}

### Vassiliev invariants

 V2 and V3: (2, -3)
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 ${\displaystyle 8}$ ${\displaystyle -24}$ ${\displaystyle 32}$ ${\displaystyle {\frac {268}{3}}}$ ${\displaystyle {\frac {44}{3}}}$ ${\displaystyle -192}$ ${\displaystyle -368}$ ${\displaystyle -64}$ ${\displaystyle -56}$ ${\displaystyle {\frac {256}{3}}}$ ${\displaystyle 288}$ ${\displaystyle {\frac {2144}{3}}}$ ${\displaystyle {\frac {352}{3}}}$ ${\displaystyle {\frac {22951}{15}}}$ ${\displaystyle -{\frac {28}{5}}}$ ${\displaystyle {\frac {29764}{45}}}$ ${\displaystyle {\frac {137}{9}}}$ ${\displaystyle {\frac {1351}{15}}}$

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 5 2. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-5-4-3-2-10χ
-1     11
-3    110
-5   1  1
-7   1  1
-9 11   0
-11      0
-131     -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle {\mathbb {Z} }}$