K11a147

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K11a146.gif

K11a146

K11a148.gif

K11a148

Contents

K11a147.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a147 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,4,11,3 X16,6,17,5 X20,8,21,7 X2,10,3,9 X22,11,1,12 X18,13,19,14 X8,16,9,15 X12,17,13,18 X6,20,7,19 X14,21,15,22
Gauss code 1, -5, 2, -1, 3, -10, 4, -8, 5, -2, 6, -9, 7, -11, 8, -3, 9, -7, 10, -4, 11, -6
Dowker-Thistlethwaite code 4 10 16 20 2 22 18 8 12 6 14
A Braid Representative
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A Morse Link Presentation K11a147 ML.gif

Three dimensional invariants

Symmetry type Chiral
Unknotting number \{1,2\}
3-genus 4
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a147/ThurstonBennequinNumber
Hyperbolic Volume 17.2353
A-Polynomial See Data:K11a147/A-polynomial

[edit Notes for K11a147's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a147's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^4+6 t^3-17 t^2+32 t-39+32 t^{-1} -17 t^{-2} +6 t^{-3} - t^{-4}
Conway polynomial -z^8-2 z^6-z^4+2 z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 151, 2 }
Jones polynomial -q^8+4 q^7-10 q^6+16 q^5-21 q^4+25 q^3-24 q^2+21 q-15+9 q^{-1} -4 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) -z^8 a^{-2} -5 z^6 a^{-2} +2 z^6 a^{-4} +z^6-10 z^4 a^{-2} +7 z^4 a^{-4} -z^4 a^{-6} +3 z^4-8 z^2 a^{-2} +9 z^2 a^{-4} -2 z^2 a^{-6} +3 z^2-2 a^{-2} +4 a^{-4} -2 a^{-6} +1
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10} a^{-4} +6 z^9 a^{-1} +13 z^9 a^{-3} +7 z^9 a^{-5} +15 z^8 a^{-2} +19 z^8 a^{-4} +11 z^8 a^{-6} +7 z^8+4 a z^7-4 z^7 a^{-1} -13 z^7 a^{-3} +4 z^7 a^{-5} +9 z^7 a^{-7} +a^2 z^6-41 z^6 a^{-2} -45 z^6 a^{-4} -16 z^6 a^{-6} +4 z^6 a^{-8} -15 z^6-9 a z^5-12 z^5 a^{-1} -15 z^5 a^{-3} -27 z^5 a^{-5} -14 z^5 a^{-7} +z^5 a^{-9} -2 a^2 z^4+32 z^4 a^{-2} +34 z^4 a^{-4} +10 z^4 a^{-6} -4 z^4 a^{-8} +10 z^4+6 a z^3+12 z^3 a^{-1} +19 z^3 a^{-3} +24 z^3 a^{-5} +10 z^3 a^{-7} -z^3 a^{-9} +a^2 z^2-12 z^2 a^{-2} -14 z^2 a^{-4} -5 z^2 a^{-6} +z^2 a^{-8} -3 z^2-a z-3 z a^{-1} -5 z a^{-3} -7 z a^{-5} -4 z a^{-7} +2 a^{-2} +4 a^{-4} +2 a^{-6} +1
The A2 invariant q^8-2 q^6+3 q^4-2 q^2-1+4 q^{-2} -5 q^{-4} +5 q^{-6} -2 q^{-8} +2 q^{-10} +3 q^{-12} -3 q^{-14} +4 q^{-16} -3 q^{-18} - q^{-20} + q^{-22} - q^{-24}
The G2 invariant q^{46}-3 q^{44}+8 q^{42}-16 q^{40}+22 q^{38}-25 q^{36}+14 q^{34}+18 q^{32}-66 q^{30}+128 q^{28}-176 q^{26}+175 q^{24}-101 q^{22}-63 q^{20}+289 q^{18}-494 q^{16}+595 q^{14}-503 q^{12}+187 q^{10}+276 q^8-746 q^6+1033 q^4-982 q^2+583+52 q^{-2} -689 q^{-4} +1072 q^{-6} -1041 q^{-8} +605 q^{-10} +49 q^{-12} -639 q^{-14} +889 q^{-16} -697 q^{-18} +143 q^{-20} +532 q^{-22} -1009 q^{-24} +1072 q^{-26} -654 q^{-28} -117 q^{-30} +932 q^{-32} -1485 q^{-34} +1542 q^{-36} -1054 q^{-38} +200 q^{-40} +732 q^{-42} -1392 q^{-44} +1563 q^{-46} -1188 q^{-48} +435 q^{-50} +387 q^{-52} -955 q^{-54} +1059 q^{-56} -686 q^{-58} +53 q^{-60} +563 q^{-62} -860 q^{-64} +723 q^{-66} -231 q^{-68} -416 q^{-70} +919 q^{-72} -1075 q^{-74} +828 q^{-76} -285 q^{-78} -335 q^{-80} +801 q^{-82} -973 q^{-84} +831 q^{-86} -473 q^{-88} +42 q^{-90} +307 q^{-92} -501 q^{-94} +504 q^{-96} -370 q^{-98} +187 q^{-100} -9 q^{-102} -105 q^{-104} +149 q^{-106} -143 q^{-108} +99 q^{-110} -49 q^{-112} +12 q^{-114} +12 q^{-116} -19 q^{-118} +17 q^{-120} -13 q^{-122} +7 q^{-124} -3 q^{-126} + q^{-128}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a322,}

Vassiliev invariants

V2 and V3: (2, 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
8 32 32 \frac{364}{3} \frac{68}{3} 256 \frac{1568}{3} \frac{128}{3} 128 \frac{256}{3} 512 \frac{2912}{3} \frac{544}{3} \frac{35911}{15} -\frac{3964}{15} \frac{55564}{45} \frac{905}{9} \frac{2551}{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 K11a147. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-4-3-2-101234567χ
17           1-1
15          3 3
13         71 -6
11        93  6
9       127   -5
7      139    4
5     1112     1
3    1013      -3
1   612       6
-1  39        -6
-3 16         5
-5 3          -3
-71           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-4 {\mathbb Z}
r=-3 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-1 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=0 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{10}
r=1 {\mathbb Z}^{13}\oplus{\mathbb Z}_2^{11} {\mathbb Z}^{11}
r=2 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=4 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=6 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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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K11a146.gif

K11a146

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K11a148