K11a203

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K11a202.gif

K11a202

K11a204.gif

K11a204

Contents

K11a203.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a203 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X12,4,13,3 X16,6,17,5 X18,8,19,7 X20,10,21,9 X2,12,3,11 X22,13,1,14 X10,16,11,15 X6,18,7,17 X8,20,9,19 X14,21,15,22
Gauss code 1, -6, 2, -1, 3, -9, 4, -10, 5, -8, 6, -2, 7, -11, 8, -3, 9, -4, 10, -5, 11, -7
Dowker-Thistlethwaite code 4 12 16 18 20 2 22 10 6 8 14
A Braid Representative
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A Morse Link Presentation K11a203 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 3
3-genus 4
Bridge index 2
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a203/ThurstonBennequinNumber
Hyperbolic Volume 10.2613
A-Polynomial See Data:K11a203/A-polynomial

[edit Notes for K11a203's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a203's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+5 t^3-9 t^2+11 t-11+11 t^{-1} -9 t^{-2} +5 t^{-3} - t^{-4}
Conway polynomial -z^8-3 z^6+z^4+4 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 63, 6 }
Jones polynomial -q^{12}+3 q^{11}-5 q^{10}+7 q^9-9 q^8+9 q^7-9 q^6+8 q^5-5 q^4+4 q^3-2 q^2+q
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-6} +z^6 a^{-4} -6 z^6 a^{-6} +2 z^6 a^{-8} +5 z^4 a^{-4} -12 z^4 a^{-6} +9 z^4 a^{-8} -z^4 a^{-10} +7 z^2 a^{-4} -10 z^2 a^{-6} +10 z^2 a^{-8} -3 z^2 a^{-10} +3 a^{-4} -3 a^{-6} +2 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) z^{10} a^{-6} +z^{10} a^{-8} +2 z^9 a^{-5} +5 z^9 a^{-7} +3 z^9 a^{-9} +z^8 a^{-4} -z^8 a^{-6} +3 z^8 a^{-8} +5 z^8 a^{-10} -11 z^7 a^{-5} -22 z^7 a^{-7} -5 z^7 a^{-9} +6 z^7 a^{-11} -6 z^6 a^{-4} -14 z^6 a^{-6} -25 z^6 a^{-8} -11 z^6 a^{-10} +6 z^6 a^{-12} +18 z^5 a^{-5} +26 z^5 a^{-7} -7 z^5 a^{-9} -10 z^5 a^{-11} +5 z^5 a^{-13} +12 z^4 a^{-4} +32 z^4 a^{-6} +35 z^4 a^{-8} +5 z^4 a^{-10} -7 z^4 a^{-12} +3 z^4 a^{-14} -8 z^3 a^{-5} -7 z^3 a^{-7} +8 z^3 a^{-9} +2 z^3 a^{-11} -4 z^3 a^{-13} +z^3 a^{-15} -10 z^2 a^{-4} -20 z^2 a^{-6} -16 z^2 a^{-8} -4 z^2 a^{-10} +z^2 a^{-12} -z^2 a^{-14} -z a^{-5} +z a^{-13} +3 a^{-4} +3 a^{-6} +2 a^{-8} + a^{-10}
The A2 invariant  q^{-4} + q^{-8} + q^{-10} +2 q^{-14} - q^{-16} +2 q^{-18} - q^{-20} - q^{-22} -2 q^{-26} + q^{-28} + q^{-32} - q^{-36}
The G2 invariant  q^{-22} - q^{-24} +4 q^{-26} -5 q^{-28} +6 q^{-30} -4 q^{-32} + q^{-34} +10 q^{-36} -18 q^{-38} +27 q^{-40} -26 q^{-42} +14 q^{-44} +6 q^{-46} -28 q^{-48} +46 q^{-50} -47 q^{-52} +36 q^{-54} -8 q^{-56} -20 q^{-58} +41 q^{-60} -47 q^{-62} +36 q^{-64} -13 q^{-66} -10 q^{-68} +25 q^{-70} -25 q^{-72} +16 q^{-74} +2 q^{-76} -17 q^{-78} +25 q^{-80} -22 q^{-82} +3 q^{-84} +14 q^{-86} -36 q^{-88} +46 q^{-90} -38 q^{-92} +20 q^{-94} +6 q^{-96} -34 q^{-98} +51 q^{-100} -53 q^{-102} +34 q^{-104} -9 q^{-106} -17 q^{-108} +33 q^{-110} -32 q^{-112} +23 q^{-114} -8 q^{-116} -4 q^{-118} +11 q^{-120} -11 q^{-122} +2 q^{-124} +5 q^{-126} -9 q^{-128} +11 q^{-130} -5 q^{-132} - q^{-134} +7 q^{-136} -13 q^{-138} +18 q^{-140} -19 q^{-142} +15 q^{-144} -7 q^{-146} -6 q^{-148} +17 q^{-150} -25 q^{-152} +26 q^{-154} -21 q^{-156} +12 q^{-158} + q^{-160} -16 q^{-162} +22 q^{-164} -24 q^{-166} +20 q^{-168} -11 q^{-170} +2 q^{-172} +6 q^{-174} -12 q^{-176} +14 q^{-178} -11 q^{-180} +8 q^{-182} -2 q^{-184} - q^{-186} +2 q^{-188} -4 q^{-190} +3 q^{-192} -2 q^{-194} + q^{-196}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (4, 9)
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
16 72 128 \frac{1160}{3} \frac{160}{3} 1152 2256 320 328 \frac{2048}{3} 2592 \frac{18560}{3} \frac{2560}{3} \frac{203222}{15} -\frac{1168}{15} \frac{238448}{45} \frac{2218}{9} \frac{10022}{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=6 is the signature of K11a203. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
25           1-1
23          2 2
21         31 -2
19        42  2
17       53   -2
15      44    0
13     55     0
11    34      -1
9   25       3
7  23        -1
5 13         2
3 1          -1
11           1
Integral Khovanov Homology

(db, data source)

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

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