9 39

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Knot presentations

 Planar diagram presentation X1627 X3,11,4,10 X7,18,8,1 X17,13,18,12 X9,17,10,16 X5,15,6,14 X15,5,16,4 X11,3,12,2 X13,9,14,8 Gauss code -1, 8, -2, 7, -6, 1, -3, 9, -5, 2, -8, 4, -9, 6, -7, 5, -4, 3 Dowker-Thistlethwaite code 6 10 14 18 16 2 8 4 12 Conway Notation [2:2:20]

Minimum Braid Representative A Morse Link Presentation An Arc Presentation

Length is 12, width is 5,

Braid index is 5

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

Three dimensional invariants

 Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 Super bridge index ${\displaystyle \{4,6\}}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-10] Hyperbolic Volume 12.8103 A-Polynomial See Data:9 39/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 ${\displaystyle -3t^{2}+14t-21+14t^{-1}-3t^{-2}}$ Conway polynomial ${\displaystyle -3z^{4}+2z^{2}+1}$ 2nd Alexander ideal (db, data sources) ${\displaystyle \{1\}}$ Determinant and Signature { 55, 2 } Jones polynomial ${\displaystyle -q^{8}+3q^{7}-6q^{6}+8q^{5}-9q^{4}+10q^{3}-8q^{2}+6q-3+q^{-1}}$ HOMFLY-PT polynomial (db, data sources) ${\displaystyle -z^{4}a^{-2}-2z^{4}a^{-4}+z^{2}a^{-2}-3z^{2}a^{-4}+3z^{2}a^{-6}+z^{2}+2a^{-2}-2a^{-4}+2a^{-6}-a^{-8}}$ Kauffman polynomial (db, data sources) ${\displaystyle 2z^{8}a^{-4}+2z^{8}a^{-6}+5z^{7}a^{-3}+9z^{7}a^{-5}+4z^{7}a^{-7}+5z^{6}a^{-2}+5z^{6}a^{-4}+3z^{6}a^{-6}+3z^{6}a^{-8}+3z^{5}a^{-1}-7z^{5}a^{-3}-18z^{5}a^{-5}-7z^{5}a^{-7}+z^{5}a^{-9}-7z^{4}a^{-2}-15z^{4}a^{-4}-13z^{4}a^{-6}-6z^{4}a^{-8}+z^{4}-3z^{3}a^{-1}+5z^{3}a^{-3}+12z^{3}a^{-5}+2z^{3}a^{-7}-2z^{3}a^{-9}+5z^{2}a^{-2}+12z^{2}a^{-4}+9z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}-za^{-3}-3za^{-5}-za^{-7}+za^{-9}-2a^{-2}-2a^{-4}-2a^{-6}-a^{-8}}$ The A2 invariant ${\displaystyle q^{4}-q^{2}-1+3q^{-2}-q^{-4}+2q^{-6}+q^{-8}-q^{-10}+q^{-12}-2q^{-14}+2q^{-16}-q^{-20}+2q^{-22}-q^{-24}-q^{-26}}$ The G2 invariant ${\displaystyle q^{18}-2q^{16}+4q^{14}-6q^{12}+5q^{10}-3q^{8}-2q^{6}+12q^{4}-19q^{2}+28-30q^{-2}+21q^{-4}-3q^{-6}-27q^{-8}+58q^{-10}-76q^{-12}+73q^{-14}-45q^{-16}-6q^{-18}+63q^{-20}-97q^{-22}+101q^{-24}-61q^{-26}+2q^{-28}+53q^{-30}-80q^{-32}+65q^{-34}-12q^{-36}-45q^{-38}+87q^{-40}-83q^{-42}+36q^{-44}+37q^{-46}-103q^{-48}+134q^{-50}-123q^{-52}+66q^{-54}+10q^{-56}-84q^{-58}+131q^{-60}-134q^{-62}+95q^{-64}-29q^{-66}-43q^{-68}+87q^{-70}-93q^{-72}+59q^{-74}-52q^{-78}+80q^{-80}-61q^{-82}+8q^{-84}+57q^{-86}-100q^{-88}+103q^{-90}-65q^{-92}-q^{-94}+60q^{-96}-93q^{-98}+95q^{-100}-63q^{-102}+19q^{-104}+19q^{-106}-45q^{-108}+45q^{-110}-33q^{-112}+17q^{-114}-3q^{-116}-6q^{-118}+8q^{-120}-8q^{-122}+5q^{-124}-2q^{-126}+q^{-128}}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n162,}

Same Jones Polynomial (up to mirroring, ${\displaystyle q\leftrightarrow q^{-1}}$): {K11n11, K11n112,}

Vassiliev invariants

 V2 and V3: (2, 4)
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 ${\displaystyle 8}$ ${\displaystyle 32}$ ${\displaystyle 32}$ ${\displaystyle {\frac {460}{3}}}$ ${\displaystyle {\frac {116}{3}}}$ ${\displaystyle 256}$ ${\displaystyle {\frac {2144}{3}}}$ ${\displaystyle {\frac {320}{3}}}$ ${\displaystyle 192}$ ${\displaystyle {\frac {256}{3}}}$ ${\displaystyle 512}$ ${\displaystyle {\frac {3680}{3}}}$ ${\displaystyle {\frac {928}{3}}}$ ${\displaystyle {\frac {49111}{15}}}$ ${\displaystyle -{\frac {8524}{15}}}$ ${\displaystyle {\frac {97084}{45}}}$ ${\displaystyle {\frac {857}{9}}}$ ${\displaystyle {\frac {5191}{15}}}$

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 9 39. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.
 \ r \ j \
-2-101234567χ
17         1-1
15        2 2
13       41 -3
11      42  2
9     54   -1
7    54    1
5   35     2
3  35      -2
1 14       3
-1 2        -2
-31         1
Integral Khovanov Homology
 ${\displaystyle \dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}}$ ${\displaystyle i=1}$ ${\displaystyle i=3}$ ${\displaystyle r=-2}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=-1}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$ ${\displaystyle r=0}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=1}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}}$ ${\displaystyle {\mathbb {Z} }^{3}}$ ${\displaystyle r=2}$ ${\displaystyle {\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=3}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{5}}$ ${\displaystyle {\mathbb {Z} }^{5}}$ ${\displaystyle r=4}$ ${\displaystyle {\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=5}$ ${\displaystyle {\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{4}}$ ${\displaystyle {\mathbb {Z} }^{4}}$ ${\displaystyle r=6}$ ${\displaystyle {\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}}$ ${\displaystyle {\mathbb {Z} }^{2}}$ ${\displaystyle r=7}$ ${\displaystyle {\mathbb {Z} }_{2}}$ ${\displaystyle {\mathbb {Z} }}$