K11a24

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K11a23.gif

K11a23

K11a25.gif

K11a25

Contents

K11a24.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a24 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X12,5,13,6 X2837 X18,10,19,9 X16,11,17,12 X6,13,7,14 X20,15,21,16 X22,18,1,17 X14,19,15,20 X10,22,11,21
Gauss code 1, -4, 2, -1, 3, -7, 4, -2, 5, -11, 6, -3, 7, -10, 8, -6, 9, -5, 10, -8, 11, -9
Dowker-Thistlethwaite code 4 8 12 2 18 16 6 20 22 14 10
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
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A Morse Link Presentation K11a24 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:K11a24/ThurstonBennequinNumber
Hyperbolic Volume 17.9006
A-Polynomial See Data:K11a24/A-polynomial

[edit Notes for K11a24's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 4
Rasmussen s-Invariant 0

[edit Notes for K11a24's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-6 t^3+18 t^2-33 t+41-33 t^{-1} +18 t^{-2} -6 t^{-3} + t^{-4}
Conway polynomial z^8+2 z^6+2 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 157, 0 }
Jones polynomial q^6-4 q^5+9 q^4-16 q^3+22 q^2-25 q+26-22 q^{-1} +17 q^{-2} -10 q^{-3} +4 q^{-4} - q^{-5}
HOMFLY-PT polynomial (db, data sources) z^8-a^2 z^6-2 z^6 a^{-2} +5 z^6-3 a^2 z^4-7 z^4 a^{-2} +z^4 a^{-4} +11 z^4-4 a^2 z^2-9 z^2 a^{-2} +2 z^2 a^{-4} +12 z^2-2 a^2-4 a^{-2} + a^{-4} +6
Kauffman polynomial (db, data sources) 2 z^{10} a^{-2} +2 z^{10}+8 a z^9+14 z^9 a^{-1} +6 z^9 a^{-3} +12 a^2 z^8+18 z^8 a^{-2} +7 z^8 a^{-4} +23 z^8+9 a^3 z^7+2 a z^7-12 z^7 a^{-1} -z^7 a^{-3} +4 z^7 a^{-5} +4 a^4 z^6-19 a^2 z^6-50 z^6 a^{-2} -14 z^6 a^{-4} +z^6 a^{-6} -58 z^6+a^5 z^5-13 a^3 z^5-26 a z^5-24 z^5 a^{-1} -21 z^5 a^{-3} -9 z^5 a^{-5} -4 a^4 z^4+15 a^2 z^4+42 z^4 a^{-2} +9 z^4 a^{-4} -2 z^4 a^{-6} +50 z^4-a^5 z^3+9 a^3 z^3+25 a z^3+30 z^3 a^{-1} +22 z^3 a^{-3} +7 z^3 a^{-5} +a^4 z^2-8 a^2 z^2-18 z^2 a^{-2} -3 z^2 a^{-4} +z^2 a^{-6} -23 z^2-3 a^3 z-8 a z-10 z a^{-1} -7 z a^{-3} -2 z a^{-5} +2 a^2+4 a^{-2} + a^{-4} +6
The A2 invariant -q^{14}+2 q^{12}-4 q^{10}+2 q^8+q^6-3 q^4+7 q^2-3+5 q^{-2} - q^{-4} -2 q^{-6} +3 q^{-8} -5 q^{-10} +2 q^{-12} - q^{-16} + q^{-18}
The G2 invariant q^{80}-3 q^{78}+7 q^{76}-13 q^{74}+17 q^{72}-19 q^{70}+12 q^{68}+11 q^{66}-46 q^{64}+94 q^{62}-139 q^{60}+154 q^{58}-123 q^{56}+20 q^{54}+160 q^{52}-369 q^{50}+543 q^{48}-584 q^{46}+408 q^{44}-30 q^{42}-482 q^{40}+949 q^{38}-1167 q^{36}+1001 q^{34}-454 q^{32}-314 q^{30}+1007 q^{28}-1337 q^{26}+1155 q^{24}-520 q^{22}-297 q^{20}+930 q^{18}-1100 q^{16}+723 q^{14}+43 q^{12}-830 q^{10}+1288 q^8-1170 q^6+494 q^4+492 q^2-1389+1846 q^{-2} -1655 q^{-4} +877 q^{-6} +226 q^{-8} -1251 q^{-10} +1840 q^{-12} -1780 q^{-14} +1119 q^{-16} -121 q^{-18} -823 q^{-20} +1338 q^{-22} -1252 q^{-24} +639 q^{-26} +208 q^{-28} -899 q^{-30} +1117 q^{-32} -782 q^{-34} +40 q^{-36} +763 q^{-38} -1275 q^{-40} +1280 q^{-42} -790 q^{-44} +9 q^{-46} +733 q^{-48} -1173 q^{-50} +1190 q^{-52} -823 q^{-54} +270 q^{-56} +253 q^{-58} -591 q^{-60} +671 q^{-62} -539 q^{-64} +300 q^{-66} -52 q^{-68} -120 q^{-70} +194 q^{-72} -188 q^{-74} +134 q^{-76} -66 q^{-78} +16 q^{-80} +15 q^{-82} -26 q^{-84} +22 q^{-86} -16 q^{-88} +8 q^{-90} -3 q^{-92} + q^{-94}

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a26, K11a315,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11a26, K11a315,}

Vassiliev invariants

V2 and V3: (1, -1)
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
4 -8 8 -\frac{34}{3} -\frac{38}{3} -32 -\frac{80}{3} \frac{64}{3} -8 \frac{32}{3} 32 -\frac{136}{3} -\frac{152}{3} \frac{751}{30} -\frac{902}{15} \frac{3302}{45} -\frac{175}{18} \frac{271}{30}

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=0 is the signature of K11a24. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-5-4-3-2-10123456χ
13           11
11          3 -3
9         61 5
7        103  -7
5       126   6
3      1310    -3
1     1312     1
-1    1014      4
-3   712       -5
-5  310        7
-7 17         -6
-9 3          3
-111           -1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=-1 i=1
r=-5 {\mathbb Z}
r=-4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-3 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=-2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=-1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=0 {\mathbb Z}^{14}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{13}
r=1 {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{13} {\mathbb Z}^{13}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{12} {\mathbb Z}^{12}
r=3 {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{6} {\mathbb Z}^{6}
r=5 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
r=6 {\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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K11a23.gif

K11a23

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K11a25