K11n41

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K11n40.gif

K11n40

K11n42.gif

K11n42

Contents

K11n41.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n41 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,13,6,12 X2837 X9,18,10,19 X11,17,12,16 X13,7,14,6 X15,21,16,20 X17,22,18,1 X19,15,20,14 X21,10,22,11
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, -5, 11, -6, 3, -7, 10, -8, 6, -9, 5, -10, 8, -11, 9
Dowker-Thistlethwaite code 4 8 -12 2 -18 -16 -6 -20 -22 -14 -10
A Braid Representative
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A Morse Link Presentation K11n41 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n41/ThurstonBennequinNumber
Hyperbolic Volume 13.7623
A-Polynomial See Data:K11n41/A-polynomial

[edit Notes for K11n41's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n41's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-4 t^3+8 t^2-9 t+9-9 t^{-1} +8 t^{-2} -4 t^{-3} + t^{-4}
Conway polynomial z^8+4 z^6+4 z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 53, 4 }
Jones polynomial q^8-4 q^7+6 q^6-8 q^5+9 q^4-8 q^3+8 q^2-5 q+3- q^{-1}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -4 z^4 a^{-2} +13 z^4 a^{-4} -5 z^4 a^{-6} -4 z^2 a^{-2} +14 z^2 a^{-4} -8 z^2 a^{-6} +z^2 a^{-8} - a^{-2} +6 a^{-4} -5 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) 2 z^9 a^{-3} +2 z^9 a^{-5} +3 z^8 a^{-2} +9 z^8 a^{-4} +6 z^8 a^{-6} +z^7 a^{-1} -3 z^7 a^{-3} +2 z^7 a^{-5} +6 z^7 a^{-7} -13 z^6 a^{-2} -36 z^6 a^{-4} -21 z^6 a^{-6} +2 z^6 a^{-8} -4 z^5 a^{-1} -12 z^5 a^{-3} -28 z^5 a^{-5} -20 z^5 a^{-7} +17 z^4 a^{-2} +42 z^4 a^{-4} +24 z^4 a^{-6} -z^4 a^{-8} +5 z^3 a^{-1} +20 z^3 a^{-3} +35 z^3 a^{-5} +24 z^3 a^{-7} +4 z^3 a^{-9} -8 z^2 a^{-2} -22 z^2 a^{-4} -15 z^2 a^{-6} +z^2 a^{-10} -2 z a^{-1} -7 z a^{-3} -13 z a^{-5} -10 z a^{-7} -2 z a^{-9} + a^{-2} +6 a^{-4} +5 a^{-6} + a^{-8}
The A2 invariant -q^2+1- q^{-2} + q^{-4} + q^{-6} + q^{-8} +4 q^{-10} - q^{-12} +3 q^{-14} -2 q^{-16} - q^{-18} - q^{-20} -2 q^{-22} + q^{-24} - q^{-26} + q^{-28}
The G2 invariant Data:K11n41/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n47,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n47,}

Vassiliev invariants

V2 and V3: (3, 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
12 32 72 110 2 384 \frac{1472}{3} \frac{128}{3} 64 288 512 1320 24 \frac{21471}{10} \frac{614}{15} \frac{3194}{5} \frac{43}{2} \frac{511}{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=4 is the signature of K11n41. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-10123456χ
17         11
15        3 -3
13       31 2
11      53  -2
9     43   1
7    45    1
5   44     0
3  25      3
1 13       -2
-1 2        2
-31         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-3 {\mathbb Z}
r=-2 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11n40.gif

K11n40

K11n42.gif

K11n42