K11a148

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K11a147.gif

K11a147

K11a149.gif

K11a149

Contents

K11a148.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a148 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X16,6,17,5 X20,8,21,7 X12,10,13,9 X2,11,3,12 X18,14,19,13 X6,16,7,15 X22,18,1,17 X8,20,9,19 X14,22,15,21
Gauss code 1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -5, 7, -11, 8, -3, 9, -7, 10, -4, 11, -9
Dowker-Thistlethwaite code 4 10 16 20 12 2 18 6 22 8 14
A Braid Representative
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A Morse Link Presentation K11a148 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 2
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a148/ThurstonBennequinNumber
Hyperbolic Volume 15.7728
A-Polynomial See Data:K11a148/A-polynomial

[edit Notes for K11a148's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11a148's four dimensional invariants]

Polynomial invariants

Alexander polynomial -7 t^2+29 t-43+29 t^{-1} -7 t^{-2}
Conway polynomial -7 z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 115, 2 }
Jones polynomial -q^{10}+4 q^9-8 q^8+12 q^7-16 q^6+18 q^5-18 q^4+16 q^3-11 q^2+7 q-3+ q^{-1}
HOMFLY-PT polynomial (db, data sources) -z^4 a^{-2} -3 z^4 a^{-4} -3 z^4 a^{-6} +2 z^2 a^{-2} -2 z^2 a^{-4} -4 z^2 a^{-6} +4 z^2 a^{-8} +z^2+2 a^{-2} -3 a^{-6} +3 a^{-8} - a^{-10}
Kauffman polynomial (db, data sources) 2 z^{10} a^{-6} +2 z^{10} a^{-8} +6 z^9 a^{-5} +11 z^9 a^{-7} +5 z^9 a^{-9} +8 z^8 a^{-4} +10 z^8 a^{-6} +6 z^8 a^{-8} +4 z^8 a^{-10} +8 z^7 a^{-3} -5 z^7 a^{-5} -28 z^7 a^{-7} -14 z^7 a^{-9} +z^7 a^{-11} +6 z^6 a^{-2} -9 z^6 a^{-4} -38 z^6 a^{-6} -37 z^6 a^{-8} -14 z^6 a^{-10} +3 z^5 a^{-1} -10 z^5 a^{-3} -7 z^5 a^{-5} +14 z^5 a^{-7} +5 z^5 a^{-9} -3 z^5 a^{-11} -7 z^4 a^{-2} -z^4 a^{-4} +35 z^4 a^{-6} +42 z^4 a^{-8} +14 z^4 a^{-10} +z^4-2 z^3 a^{-1} +6 z^3 a^{-3} +5 z^3 a^{-5} +z^3 a^{-7} +7 z^3 a^{-9} +3 z^3 a^{-11} +6 z^2 a^{-2} +4 z^2 a^{-4} -16 z^2 a^{-6} -17 z^2 a^{-8} -4 z^2 a^{-10} -z^2-z a^{-5} -2 z a^{-7} -2 z a^{-9} -z a^{-11} -2 a^{-2} +3 a^{-6} +3 a^{-8} + a^{-10}
The A2 invariant Data:K11a148/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a148/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

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

Vassiliev invariants

V2 and V3: (1, 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
4 16 8 \frac{398}{3} \frac{178}{3} 64 \frac{2080}{3} \frac{448}{3} 272 \frac{32}{3} 128 \frac{1592}{3} \frac{712}{3} \frac{83071}{30} -\frac{12062}{15} \frac{112382}{45} \frac{3137}{18} \frac{11071}{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=2 is the signature of K11a148. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-2-10123456789χ
21           1-1
19          3 3
17         51 -4
15        73  4
13       95   -4
11      97    2
9     99     0
7    79      -2
5   49       5
3  37        -4
1 15         4
-1 2          -2
-31           1
Integral Khovanov Homology

(db, data source)

  
\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} i=1 i=3
r=-2 {\mathbb Z}
r=-1 {\mathbb Z}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=0 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{3}
r=1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{4} {\mathbb Z}^{4}
r=2 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=3 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=4 {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=5 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=6 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{7}
r=7 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=8 {\mathbb Z}\oplus{\mathbb Z}_2^{3} {\mathbb Z}^{3}
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).

See/edit the Hoste-Thistlethwaite_Splice_Base (expert).

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K11a147.gif

K11a147

K11a149.gif

K11a149