K11a104

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K11a103.gif

K11a103

K11a105.gif

K11a105

Contents

K11a104.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a104 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X14,5,15,6 X12,8,13,7 X20,10,21,9 X2,11,3,12 X16,13,17,14 X6,15,7,16 X22,18,1,17 X8,20,9,19 X18,22,19,21
Gauss code 1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -4, 7, -3, 8, -7, 9, -11, 10, -5, 11, -9
Dowker-Thistlethwaite code 4 10 14 12 20 2 16 6 22 8 18
A Braid Representative
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A Morse Link Presentation K11a104 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a104/ThurstonBennequinNumber
Hyperbolic Volume 15.7183
A-Polynomial See Data:K11a104/A-polynomial

[edit Notes for K11a104's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 1
Rasmussen s-Invariant 0

[edit Notes for K11a104's four dimensional invariants]

Polynomial invariants

Alexander polynomial -2 t^3+12 t^2-29 t+39-29 t^{-1} +12 t^{-2} -2 t^{-3}
Conway polynomial -2 z^6+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 125, 0 }
Jones polynomial q^6-4 q^5+8 q^4-13 q^3+18 q^2-20 q+20-17 q^{-1} +13 q^{-2} -7 q^{-3} +3 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-2} -z^6+2 a^2 z^4-2 z^4 a^{-2} +z^4 a^{-4} -z^4-a^4 z^2+3 a^2 z^2-2 z^2 a^{-2} +z^2 a^{-4} -a^4+2 a^2
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10}+4 a z^9+8 z^9 a^{-1} +4 z^9 a^{-3} +6 a^2 z^8+14 z^8 a^{-2} +6 z^8 a^{-4} +14 z^8+5 a^3 z^7+4 a z^7-3 z^7 a^{-1} +2 z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-6 a^2 z^6-36 z^6 a^{-2} -13 z^6 a^{-4} +z^6 a^{-6} -31 z^6+a^5 z^5-6 a^3 z^5-16 a z^5-21 z^5 a^{-1} -22 z^5 a^{-3} -10 z^5 a^{-5} -5 a^4 z^4+a^2 z^4+26 z^4 a^{-2} +7 z^4 a^{-4} -2 z^4 a^{-6} +23 z^4-2 a^5 z^3+2 a^3 z^3+14 a z^3+21 z^3 a^{-1} +18 z^3 a^{-3} +7 z^3 a^{-5} +3 a^4 z^2+3 a^2 z^2-7 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} -5 z^2+a^5 z-4 a z-6 z a^{-1} -4 z a^{-3} -z a^{-5} -a^4-2 a^2
The A2 invariant Data:K11a104/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a104/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a67, K11a168,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a168,}

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
4 -8 8 \frac{62}{3} \frac{10}{3} -32 -\frac{80}{3} \frac{64}{3} -8 \frac{32}{3} 32 \frac{248}{3} \frac{40}{3} \frac{6031}{30} \frac{1738}{15} -\frac{1258}{45} \frac{305}{18} -\frac{1169}{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 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=0 is the signature of K11a104. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
13           11
11          3 -3
9         51 4
7        83  -5
5       105   5
3      108    -2
1     1010     0
-1    811      3
-3   59       -4
-5  28        6
-7 15         -4
-9 2          2
-111           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=0 {\mathbb Z}^{11}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=2 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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K11a103.gif

K11a103

K11a105.gif

K11a105