K11a9

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K11a8.gif

K11a8

K11a10.gif

K11a10

Contents

K11a9.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a9 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8394 X10,6,11,5 X16,7,17,8 X2,9,3,10 X18,12,19,11 X20,14,21,13 X22,16,1,15 X6,17,7,18 X12,20,13,19 X14,22,15,21
Gauss code 1, -5, 2, -1, 3, -9, 4, -2, 5, -3, 6, -10, 7, -11, 8, -4, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code 4 8 10 16 2 18 20 22 6 12 14
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart3.gifBraidPart0.gifBraidPart3.gifBraidPart0.gif
BraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart4.gifBraidPart1.gifBraidPart4.gifBraidPart1.gif
BraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart0.gifBraidPart2.gif
A Morse Link Presentation K11a9 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 2
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a9/ThurstonBennequinNumber
Hyperbolic Volume 11.385
A-Polynomial See Data:K11a9/A-polynomial

[edit Notes for K11a9'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 K11a9's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-5 t^3+10 t^2-11 t+11-11 t^{-1} +10 t^{-2} -5 t^{-3} + t^{-4}
Conway polynomial z^8+3 z^6+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 65, 4 }
Jones polynomial -q^9+3 q^8-5 q^7+7 q^6-9 q^5+10 q^4-9 q^3+8 q^2-6 q+4-2 q^{-1} + q^{-2}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -2 z^6 a^{-2} +6 z^6 a^{-4} -z^6 a^{-6} -10 z^4 a^{-2} +13 z^4 a^{-4} -4 z^4 a^{-6} +z^4-14 z^2 a^{-2} +14 z^2 a^{-4} -4 z^2 a^{-6} +4 z^2-6 a^{-2} +6 a^{-4} -2 a^{-6} +3
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +2 z^9 a^{-1} +6 z^9 a^{-3} +4 z^9 a^{-5} +5 z^8 a^{-4} +6 z^8 a^{-6} +z^8-11 z^7 a^{-1} -28 z^7 a^{-3} -11 z^7 a^{-5} +6 z^7 a^{-7} -20 z^6 a^{-2} -36 z^6 a^{-4} -16 z^6 a^{-6} +6 z^6 a^{-8} -6 z^6+19 z^5 a^{-1} +38 z^5 a^{-3} +5 z^5 a^{-5} -9 z^5 a^{-7} +5 z^5 a^{-9} +44 z^4 a^{-2} +54 z^4 a^{-4} +12 z^4 a^{-6} -7 z^4 a^{-8} +3 z^4 a^{-10} +12 z^4-12 z^3 a^{-1} -17 z^3 a^{-3} -z^3 a^{-5} -z^3 a^{-7} -4 z^3 a^{-9} +z^3 a^{-11} -30 z^2 a^{-2} -29 z^2 a^{-4} -7 z^2 a^{-6} +z^2 a^{-8} -z^2 a^{-10} -10 z^2+2 z a^{-1} +3 z a^{-3} +z a^{-5} +z a^{-7} +z a^{-9} +6 a^{-2} +6 a^{-4} +2 a^{-6} +3
The A2 invariant q^6+q^4+1- q^{-2} - q^{-6} - q^{-8} +2 q^{-10} - q^{-12} +3 q^{-14} - q^{-22} + q^{-24} - q^{-26}
The G2 invariant q^{26}-q^{24}+4 q^{22}-5 q^{20}+6 q^{18}-5 q^{16}+q^{14}+10 q^{12}-19 q^{10}+30 q^8-29 q^6+19 q^4+4 q^2-31+54 q^{-2} -58 q^{-4} +43 q^{-6} -13 q^{-8} -24 q^{-10} +50 q^{-12} -58 q^{-14} +42 q^{-16} -14 q^{-18} -18 q^{-20} +34 q^{-22} -35 q^{-24} +12 q^{-26} +13 q^{-28} -30 q^{-30} +36 q^{-32} -26 q^{-34} + q^{-36} +26 q^{-38} -50 q^{-40} +57 q^{-42} -47 q^{-44} +21 q^{-46} +16 q^{-48} -43 q^{-50} +60 q^{-52} -55 q^{-54} +38 q^{-56} -5 q^{-58} -24 q^{-60} +39 q^{-62} -35 q^{-64} +19 q^{-66} +5 q^{-68} -15 q^{-70} +19 q^{-72} -7 q^{-74} -8 q^{-76} +18 q^{-78} -21 q^{-80} +16 q^{-82} -6 q^{-84} -9 q^{-86} +17 q^{-88} -19 q^{-90} +18 q^{-92} -15 q^{-94} +9 q^{-96} -5 q^{-98} -5 q^{-100} +12 q^{-102} -22 q^{-104} +25 q^{-106} -20 q^{-108} +14 q^{-110} - q^{-112} -13 q^{-114} +21 q^{-116} -25 q^{-118} +21 q^{-120} -12 q^{-122} +2 q^{-124} +6 q^{-126} -12 q^{-128} +14 q^{-130} -11 q^{-132} +8 q^{-134} -2 q^{-136} - q^{-138} +2 q^{-140} -4 q^{-142} +3 q^{-144} -2 q^{-146} + q^{-148}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a140,}

Vassiliev invariants

V2 and V3: (0, 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
0 16 0 48 0 0 \frac{64}{3} -\frac{224}{3} 16 0 128 0 0 88 -352 240 -24 72

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 K11a9. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
19           1-1
17          2 2
15         31 -2
13        42  2
11       53   -2
9      54    1
7     45     1
5    45      -1
3   35       2
1  13        -2
-1 13         2
-3 1          -1
-51           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=3 i=5
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-1 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{4}
r=1 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=3 {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=5 {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=7 {\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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K11a8

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K11a10