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K11a78

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K11a77.gif

K11a77

K11a79.gif

K11a79

Contents

K11a78.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11a78's page at Knotilus!

Visit K11a78's page at the original Knot Atlas!



Knot presentations

Planar diagram presentation X4251 X10,3,11,4 X12,6,13,5 X14,7,15,8 X20,10,21,9 X2,11,3,12 X16,14,17,13 X6,15,7,16 X22,18,1,17 X8,20,9,19 X18,22,19,21
Gauss code 1, -6, 2, -1, 3, -8, 4, -10, 5, -2, 6, -3, 7, -4, 8, -7, 9, -11, 10, -5, 11, -9
Dowker-Thistlethwaite code 4 10 12 14 20 2 16 6 22 8 18
A Braid Representative
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A Morse Link Presentation K11a78 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number 1
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11a78/ThurstonBennequinNumber
Hyperbolic Volume 15.3316
A-Polynomial See Data:K11a78/A-polynomial

[edit Notes for K11a78's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant -2

[edit Notes for K11a78's four dimensional invariants]

Polynomial invariants

Alexander polynomial 2 t^3-12 t^2+29 t-37+29 t^{-1} -12 t^{-2} +2 t^{-3}
Conway polynomial 2 z^6-z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 123, 2 }
Jones polynomial -q^8+4 q^7-8 q^6+13 q^5-18 q^4+20 q^3-19 q^2+17 q-12+7 q^{-1} -3 q^{-2} + q^{-3}
HOMFLY-PT polynomial (db, data sources) z^6 a^{-2} +z^6 a^{-4} +z^4 a^{-2} +2 z^4 a^{-4} -z^4 a^{-6} -2 z^4+a^2 z^2+2 z^2 a^{-4} -z^2 a^{-6} -3 z^2+a^2+ a^{-2} -1
Kauffman polynomial (db, data sources) z^{10} a^{-2} +z^{10} a^{-4} +3 z^9 a^{-1} +7 z^9 a^{-3} +4 z^9 a^{-5} +10 z^8 a^{-2} +13 z^8 a^{-4} +7 z^8 a^{-6} +4 z^8+3 a z^7+3 z^7 a^{-1} +7 z^7 a^{-5} +7 z^7 a^{-7} +a^2 z^6-21 z^6 a^{-2} -24 z^6 a^{-4} -6 z^6 a^{-6} +4 z^6 a^{-8} -6 z^6-8 a z^5-19 z^5 a^{-1} -23 z^5 a^{-3} -24 z^5 a^{-5} -11 z^5 a^{-7} +z^5 a^{-9} -3 a^2 z^4+9 z^4 a^{-2} +11 z^4 a^{-4} -3 z^4 a^{-6} -6 z^4 a^{-8} -2 z^4+7 a z^3+17 z^3 a^{-1} +22 z^3 a^{-3} +19 z^3 a^{-5} +6 z^3 a^{-7} -z^3 a^{-9} +3 a^2 z^2-z^2 a^{-4} +3 z^2 a^{-6} +2 z^2 a^{-8} +5 z^2-2 a z-5 z a^{-1} -6 z a^{-3} -4 z a^{-5} -z a^{-7} -a^2- a^{-2} -1
The A2 invariant Data:K11a78/QuantumInvariant/A2/1,0
The G2 invariant Data:K11a78/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11a130,}

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

Vassiliev invariants

V2 and V3: (-1, 0)
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 0 8 \frac{34}{3} \frac{14}{3} 0 -32 -32 0 -\frac{32}{3} 0 -\frac{136}{3} -\frac{56}{3} -\frac{4111}{30} -\frac{2498}{15} \frac{2578}{45} -\frac{209}{18} \frac{1169}{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 K11a78. 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         51 -4
11        83  5
9       105   -5
7      108    2
5     910     1
3    810      -2
1   510       5
-1  27        -5
-3 15         4
-5 2          -2
-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}^{2}\oplus{\mathbb Z}_2 {\mathbb Z}
r=-2 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2} {\mathbb Z}^{2}
r=-1 {\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
r=0 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{7} {\mathbb Z}^{8}
r=1 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{9} {\mathbb Z}^{9}
r=2 {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=3 {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{10} {\mathbb Z}^{10}
r=4 {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{8} {\mathbb Z}^{8}
r=5 {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} {\mathbb Z}^{5}
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).

Back to the top.

K11a77.gif

K11a77

K11a79.gif

K11a79