K11a298

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K11a297.gif

K11a297

K11a299.gif

K11a299

Contents

K11a298.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

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Knot presentations

Planar diagram presentation X6271 X10,4,11,3 X18,6,19,5 X2837 X16,10,17,9 X20,12,21,11 X22,14,1,13 X4,16,5,15 X8,18,9,17 X14,20,15,19 X12,22,13,21
Gauss code 1, -4, 2, -8, 3, -1, 4, -9, 5, -2, 6, -11, 7, -10, 8, -5, 9, -3, 10, -6, 11, -7
Dowker-Thistlethwaite code 6 10 18 2 16 20 22 4 8 14 12
A Braid Representative
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A Morse Link Presentation K11a298 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{3,4\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a298/ThurstonBennequinNumber
Hyperbolic Volume 16.2798
A-Polynomial See Data:K11a298/A-polynomial

[edit Notes for K11a298's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -6

[edit Notes for K11a298's four dimensional invariants]

Polynomial invariants

Alexander polynomial 5 t^3-16 t^2+28 t-33+28 t^{-1} -16 t^{-2} +5 t^{-3}
Conway polynomial 5 z^6+14 z^4+9 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 131, 6 }
Jones polynomial -q^{14}+4 q^{13}-9 q^{12}+14 q^{11}-19 q^{10}+21 q^9-21 q^8+18 q^7-12 q^6+8 q^5-3 q^4+q^3
HOMFLY-PT polynomial (db, data sources) z^6 a^{-6} +3 z^6 a^{-8} +z^6 a^{-10} +3 z^4 a^{-6} +12 z^4 a^{-8} -z^4 a^{-12} +2 z^2 a^{-6} +15 z^2 a^{-8} -7 z^2 a^{-10} -z^2 a^{-12} +6 a^{-8} -6 a^{-10} + a^{-12}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-10} +2 z^{10} a^{-12} +5 z^9 a^{-9} +11 z^9 a^{-11} +6 z^9 a^{-13} +6 z^8 a^{-8} +8 z^8 a^{-10} +11 z^8 a^{-12} +9 z^8 a^{-14} +3 z^7 a^{-7} -7 z^7 a^{-9} -19 z^7 a^{-11} -z^7 a^{-13} +8 z^7 a^{-15} +z^6 a^{-6} -18 z^6 a^{-8} -29 z^6 a^{-10} -27 z^6 a^{-12} -13 z^6 a^{-14} +4 z^6 a^{-16} -7 z^5 a^{-7} -4 z^5 a^{-9} +6 z^5 a^{-11} -11 z^5 a^{-13} -13 z^5 a^{-15} +z^5 a^{-17} -3 z^4 a^{-6} +23 z^4 a^{-8} +34 z^4 a^{-10} +19 z^4 a^{-12} +6 z^4 a^{-14} -5 z^4 a^{-16} +3 z^3 a^{-7} +11 z^3 a^{-9} +8 z^3 a^{-11} +8 z^3 a^{-13} +7 z^3 a^{-15} -z^3 a^{-17} +2 z^2 a^{-6} -18 z^2 a^{-8} -21 z^2 a^{-10} -4 z^2 a^{-12} -2 z^2 a^{-14} +z^2 a^{-16} -6 z a^{-9} -5 z a^{-11} -z a^{-13} -2 z a^{-15} +6 a^{-8} +6 a^{-10} + a^{-12}
The A2 invariant Data:K11a298/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a298/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (9, 25)
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
36 200 648 1466 206 7200 \frac{35888}{3} \frac{6080}{3} 1448 7776 20000 52776 7416 \frac{1001773}{10} \frac{61562}{15} \frac{536306}{15} \frac{4403}{6} \frac{44973}{10}

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 t^rq^j are shown, along with their alternating sums \chi (fixed j, alternation over r). The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s-1, where s=6 is the signature of K11a298. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
01234567891011χ
29           1-1
27          3 3
25         61 -5
23        83  5
21       116   -5
19      108    2
17     1111     0
15    710      -3
13   511       6
11  37        -4
9  5         5
713          -2
51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=5 i=7
r=0 {\mathbb Z} {\mathbb Z}
r=1 {\mathbb Z}^{3}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=5 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=6 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=7 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=8 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=9 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=10 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=11 {\mathbb Z}_2 {\mathbb Z}

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages.

See/edit the Hoste-Thistlethwaite Knot Page master template (intermediate).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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