K11n22

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K11n21.gif

K11n21

K11n23.gif

K11n23

Contents

K11n22.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n22 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,13,6,12 X2837 X14,9,15,10 X18,11,19,12 X13,7,14,6 X20,16,21,15 X10,17,11,18 X22,20,1,19 X16,22,17,21
Gauss code 1, -4, 2, -1, -3, 7, 4, -2, 5, -9, 6, 3, -7, -5, 8, -11, 9, -6, 10, -8, 11, -10
Dowker-Thistlethwaite code 4 8 -12 2 14 18 -6 20 10 22 16
A Braid Representative
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BraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart2.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart2.gifBraidPart3.gif
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A Morse Link Presentation K11n22 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:K11n22/ThurstonBennequinNumber
Hyperbolic Volume 13.189
A-Polynomial See Data:K11n22/A-polynomial

[edit Notes for K11n22's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11n22's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^3-5 t^2+13 t-17+13 t^{-1} -5 t^{-2} + t^{-3}
Conway polynomial z^6+z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 55, 2 }
Jones polynomial q^9-3 q^8+5 q^7-8 q^6+9 q^5-9 q^4+9 q^3-6 q^2+4 q-1
HOMFLY-PT polynomial (db, data sources) z^6 a^{-4} -z^4 a^{-2} +4 z^4 a^{-4} -2 z^4 a^{-6} -z^2 a^{-2} +7 z^2 a^{-4} -5 z^2 a^{-6} +z^2 a^{-8} +4 a^{-4} -4 a^{-6} + a^{-8}
Kauffman polynomial (db, data sources) z^9 a^{-5} +z^9 a^{-7} +3 z^8 a^{-4} +6 z^8 a^{-6} +3 z^8 a^{-8} +2 z^7 a^{-3} +4 z^7 a^{-5} +5 z^7 a^{-7} +3 z^7 a^{-9} -9 z^6 a^{-4} -17 z^6 a^{-6} -7 z^6 a^{-8} +z^6 a^{-10} -4 z^5 a^{-3} -18 z^5 a^{-5} -24 z^5 a^{-7} -10 z^5 a^{-9} +4 z^4 a^{-2} +17 z^4 a^{-4} +17 z^4 a^{-6} +z^4 a^{-8} -3 z^4 a^{-10} +z^3 a^{-1} +9 z^3 a^{-3} +24 z^3 a^{-5} +25 z^3 a^{-7} +9 z^3 a^{-9} -3 z^2 a^{-2} -12 z^2 a^{-4} -11 z^2 a^{-6} +2 z^2 a^{-10} -z a^{-1} -4 z a^{-3} -10 z a^{-5} -10 z a^{-7} -3 z a^{-9} +4 a^{-4} +4 a^{-6} + a^{-8}
The A2 invariant -1+2 q^{-2} - q^{-4} + q^{-6} +3 q^{-8} +3 q^{-12} - q^{-14} - q^{-18} -3 q^{-20} + q^{-22} - q^{-24} + q^{-28}
The G2 invariant Data:K11n22/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {9_31, K11n11, K11n112, K11n127,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n127,}

Vassiliev invariants

V2 and V3: (2, 2)
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
8 16 32 \frac{124}{3} \frac{20}{3} 128 \frac{448}{3} \frac{64}{3} 48 \frac{256}{3} 128 \frac{992}{3} \frac{160}{3} \frac{7471}{15} -\frac{2044}{15} \frac{19084}{45} \frac{161}{9} \frac{991}{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 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=2 is the signature of K11n22. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-1012345678χ
19         11
17        2 -2
15       31 2
13      52  -3
11     43   1
9    55    0
7   44     0
5  25      3
3 24       -2
1 3        3
-11         -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-1 {\mathbb Z}
r=0 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}^{2}
r=1 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=6 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=7 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=8 {\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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K11n21.gif

K11n21

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K11n23