K11n105

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K11n104.gif

K11n104

K11n106.gif

K11n106

Contents

K11n105.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n105 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X14,5,15,6 X16,8,17,7 X2,10,3,9 X11,21,12,20 X13,1,14,22 X18,16,19,15 X8,18,9,17 X6,19,7,20 X21,13,22,12
Gauss code 1, -5, 2, -1, 3, -10, 4, -9, 5, -2, -6, 11, -7, -3, 8, -4, 9, -8, 10, 6, -11, 7
Dowker-Thistlethwaite code 4 10 14 16 2 -20 -22 18 8 6 -12
A Braid Representative
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A Morse Link Presentation K11n105 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n105/ThurstonBennequinNumber
Hyperbolic Volume 13.6421
A-Polynomial See Data:K11n105/A-polynomial

[edit Notes for K11n105's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n105's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+7 t^2-16 t+21-16 t^{-1} +7 t^{-2} - t^{-3}
Conway polynomial -z^6+z^4+3 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 69, 4 }
Jones polynomial -q^{11}+4 q^{10}-7 q^9+9 q^8-12 q^7+12 q^6-10 q^5+8 q^4-4 q^3+2 q^2
HOMFLY-PT polynomial (db, data sources) -z^6 a^{-6} +2 z^4 a^{-4} -3 z^4 a^{-6} +2 z^4 a^{-8} +5 z^2 a^{-4} -4 z^2 a^{-6} +3 z^2 a^{-8} -z^2 a^{-10} +3 a^{-4} -2 a^{-6}
Kauffman polynomial (db, data sources) z^9 a^{-7} +z^9 a^{-9} +2 z^8 a^{-6} +6 z^8 a^{-8} +4 z^8 a^{-10} +z^7 a^{-5} +3 z^7 a^{-7} +8 z^7 a^{-9} +6 z^7 a^{-11} -3 z^6 a^{-6} -10 z^6 a^{-8} -3 z^6 a^{-10} +4 z^6 a^{-12} +z^5 a^{-5} -5 z^5 a^{-7} -19 z^5 a^{-9} -12 z^5 a^{-11} +z^5 a^{-13} +3 z^4 a^{-4} +8 z^4 a^{-6} +8 z^4 a^{-8} -4 z^4 a^{-10} -7 z^4 a^{-12} -z^3 a^{-5} +4 z^3 a^{-7} +11 z^3 a^{-9} +5 z^3 a^{-11} -z^3 a^{-13} -6 z^2 a^{-4} -8 z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} +z^2 a^{-12} -2 z a^{-5} -z a^{-7} +z a^{-9} +3 a^{-4} +2 a^{-6}
The A2 invariant 2 q^{-6} - q^{-8} +3 q^{-10} + q^{-12} - q^{-14} +3 q^{-16} -2 q^{-18} + q^{-20} -2 q^{-22} -2 q^{-24} + q^{-26} -2 q^{-28} +2 q^{-30} + q^{-32} - q^{-34}
The G2 invariant Data:K11n105/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_78, K11n98,}

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

Vassiliev invariants

V2 and V3: (3, 5)
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 40 72 174 26 480 \frac{2512}{3} \frac{352}{3} 136 288 800 2088 312 \frac{41151}{10} -\frac{2306}{15} \frac{8914}{5} \frac{171}{2} \frac{2271}{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 K11n105. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
0123456789χ
23         1-1
21        3 3
19       41 -3
17      53  2
15     74   -3
13    55    0
11   57     2
9  35      -2
7 15       4
513        -2
32         2
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=0 {\mathbb Z}^{2} {\mathbb Z}
r=1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=3 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=5 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=9 {\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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K11n104.gif

K11n104

K11n106.gif

K11n106