9 25

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Knot presentations

 Planar diagram presentation X1425 X3849 X5,12,6,13 X9,17,10,16 X13,18,14,1 X17,14,18,15 X15,11,16,10 X11,6,12,7 X7283 Gauss code -1, 9, -2, 1, -3, 8, -9, 2, -4, 7, -8, 3, -5, 6, -7, 4, -6, 5 Dowker-Thistlethwaite code 4 8 12 2 16 6 18 10 14 Conway Notation [22,21,2]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 10, width is 5,

Braid index is 5

[{12, 4}, {3, 10}, {8, 11}, {10, 12}, {9, 5}, {4, 8}, {5, 2}, {1, 3}, {6, 9}, {2, 7}, {11, 6}, {7, 1}]

Three dimensional invariants

 Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 3 Super bridge index ${\displaystyle \{4,7\}}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-10][-1] Hyperbolic Volume 11.3903 A-Polynomial See Data:9 25/A-polynomial

Four dimensional invariants

 Smooth 4 genus ${\displaystyle 1}$ Topological 4 genus ${\displaystyle 1}$ Concordance genus ${\displaystyle 2}$ Rasmussen s-Invariant -2

Polynomial invariants

 Alexander polynomial ${\displaystyle -3t^{2}+12t-17+12t^{-1}-3t^{-2}}$ Conway polynomial ${\displaystyle 1-3z^{4}}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 47, -2 } Jones polynomial ${\displaystyle q-2+5q^{-1}-7q^{-2}+8q^{-3}-8q^{-4}+7q^{-5}-5q^{-6}+3q^{-7}-q^{-8}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -a^{8}+3z^{2}a^{6}+3a^{6}-2z^{4}a^{4}-4z^{2}a^{4}-3a^{4}-z^{4}a^{2}+a^{2}+z^{2}+1}$ Kauffman polynomial (db, data sources) ${\displaystyle z^{5}a^{9}-2z^{3}a^{9}+za^{9}+3z^{6}a^{8}-7z^{4}a^{8}+4z^{2}a^{8}-a^{8}+3z^{7}a^{7}-4z^{5}a^{7}-2z^{3}a^{7}+za^{7}+z^{8}a^{6}+6z^{6}a^{6}-18z^{4}a^{6}+13z^{2}a^{6}-3a^{6}+6z^{7}a^{5}-10z^{5}a^{5}+5z^{3}a^{5}-za^{5}+z^{8}a^{4}+6z^{6}a^{4}-15z^{4}a^{4}+13z^{2}a^{4}-3a^{4}+3z^{7}a^{3}-3z^{5}a^{3}+3z^{3}a^{3}-za^{3}+3z^{6}a^{2}-3z^{4}a^{2}+2z^{2}a^{2}-a^{2}+2z^{5}a-2z^{3}a+z^{4}-2z^{2}+1}$ The A2 invariant ${\displaystyle -q^{26}-q^{24}+2q^{22}+q^{18}+2q^{16}-2q^{14}-2q^{10}+q^{6}-q^{4}+3q^{2}+q^{-4}}$ The G2 invariant ${\displaystyle q^{128}-2q^{126}+5q^{124}-8q^{122}+7q^{120}-4q^{118}-6q^{116}+19q^{114}-29q^{112}+34q^{110}-28q^{108}+4q^{106}+23q^{104}-50q^{102}+63q^{100}-55q^{98}+26q^{96}+12q^{94}-46q^{92}+60q^{90}-48q^{88}+22q^{86}+15q^{84}-38q^{82}+41q^{80}-18q^{78}-14q^{76}+48q^{74}-60q^{72}+50q^{70}-13q^{68}-30q^{66}+68q^{64}-87q^{62}+79q^{60}-45q^{58}-6q^{56}+48q^{54}-78q^{52}+76q^{50}-50q^{48}+8q^{46}+26q^{44}-46q^{42}+38q^{40}-14q^{38}-20q^{36}+43q^{34}-43q^{32}+21q^{30}+13q^{28}-44q^{26}+63q^{24}-54q^{22}+31q^{20}-q^{18}-27q^{16}+43q^{14}-43q^{12}+34q^{10}-14q^{8}+11q^{4}-16q^{2}+15-11q^{-2}+8q^{-4}-2q^{-6}-q^{-8}+3q^{-10}-3q^{-12}+3q^{-14}-q^{-16}+q^{-18}}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n134,}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {K11n25,}

Vassiliev invariants

 V2 and V3: (0, -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 ${\displaystyle 0}$ ${\displaystyle -8}$ ${\displaystyle 0}$ ${\displaystyle 64}$ ${\displaystyle 24}$ ${\displaystyle 0}$ ${\displaystyle -{\frac {656}{3}}}$ ${\displaystyle -{\frac {128}{3}}}$ ${\displaystyle -72}$ ${\displaystyle 0}$ ${\displaystyle 32}$ ${\displaystyle 0}$ ${\displaystyle 0}$ ${\displaystyle 528}$ ${\displaystyle -{\frac {248}{3}}}$ ${\displaystyle 344}$ ${\displaystyle 48}$ ${\displaystyle 32}$

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 25. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-7-6-5-4-3-2-1012χ
3         11
1        1 -1
-1       41 3
-3      42  -2
-5     43   1
-7    44    0
-9   34     -1
-11  24      2
-13 13       -2
-15 2        2
-171         -1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=-3}$ ${\displaystyle i=-1}$ ${\displaystyle r=-7}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-6}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-5}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=-4}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=-3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$