K11n78

From Knot Atlas
Jump to: navigation, search

K11n77.gif

K11n77

K11n79.gif

K11n79

Contents

K11n78.gif
(Knotscape image)
See the full Hoste-Thistlethwaite Table of 11 Crossing Knots.

Visit K11n78 at Knotilus!



Knot presentations

Planar diagram presentation X4251 X8493 X5,15,6,14 X2837 X9,20,10,21 X11,17,12,16 X13,19,14,18 X15,7,16,6 X17,13,18,12 X19,22,20,1 X21,10,22,11
Gauss code 1, -4, 2, -1, -3, 8, 4, -2, -5, 11, -6, 9, -7, 3, -8, 6, -9, 7, -10, 5, -11, 10
Dowker-Thistlethwaite code 4 8 -14 2 -20 -16 -18 -6 -12 -22 -10
A Braid Representative
BraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart0.gif
BraidPart2.gifBraidPart3.gifBraidPart3.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart3.gifBraidPart4.gifBraidPart1.gifBraidPart0.gifBraidPart0.gifBraidPart1.gifBraidPart1.gif
BraidPart0.gifBraidPart4.gifBraidPart4.gifBraidPart1.gifBraidPart1.gifBraidPart1.gifBraidPart4.gifBraidPart0.gifBraidPart2.gifBraidPart1.gifBraidPart1.gifBraidPart2.gifBraidPart2.gif
BraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gifBraidPart0.gifBraidPart2.gifBraidPart2.gifBraidPart0.gifBraidPart0.gif
A Morse Link Presentation K11n78 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{2,3\}
3-genus 4
Bridge index 4
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n78/ThurstonBennequinNumber
Hyperbolic Volume 12.3125
A-Polynomial See Data:K11n78/A-polynomial

[edit Notes for K11n78's three dimensional invariants]

Four dimensional invariants

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

[edit Notes for K11n78's four dimensional invariants]

Polynomial invariants

Alexander polynomial t^4-3 t^3+6 t^2-8 t+9-8 t^{-1} +6 t^{-2} -3 t^{-3} + t^{-4}
Conway polynomial z^8+5 z^6+8 z^4+5 z^2+1
2nd Alexander ideal (db, data sources) \left\{t^2-t+1\right\}
Determinant and Signature { 45, 4 }
Jones polynomial q^8-4 q^7+5 q^6-7 q^5+8 q^4-6 q^3+7 q^2-4 q+2- q^{-1}
HOMFLY-PT polynomial (db, data sources) z^8 a^{-4} -z^6 a^{-2} +7 z^6 a^{-4} -z^6 a^{-6} -5 z^4 a^{-2} +19 z^4 a^{-4} -6 z^4 a^{-6} -8 z^2 a^{-2} +25 z^2 a^{-4} -13 z^2 a^{-6} +z^2 a^{-8} -4 a^{-2} +13 a^{-4} -10 a^{-6} +2 a^{-8}
Kauffman polynomial (db, data sources) z^9 a^{-3} +z^9 a^{-5} +2 z^8 a^{-2} +6 z^8 a^{-4} +4 z^8 a^{-6} +z^7 a^{-1} +z^7 a^{-3} +5 z^7 a^{-5} +5 z^7 a^{-7} -9 z^6 a^{-2} -24 z^6 a^{-4} -13 z^6 a^{-6} +2 z^6 a^{-8} -5 z^5 a^{-1} -18 z^5 a^{-3} -31 z^5 a^{-5} -18 z^5 a^{-7} +13 z^4 a^{-2} +31 z^4 a^{-4} +16 z^4 a^{-6} -2 z^4 a^{-8} +8 z^3 a^{-1} +28 z^3 a^{-3} +42 z^3 a^{-5} +26 z^3 a^{-7} +4 z^3 a^{-9} -9 z^2 a^{-2} -26 z^2 a^{-4} -17 z^2 a^{-6} +z^2 a^{-8} +z^2 a^{-10} -4 z a^{-1} -13 z a^{-3} -21 z a^{-5} -14 z a^{-7} -2 z a^{-9} +4 a^{-2} +13 a^{-4} +10 a^{-6} +2 a^{-8}
The A2 invariant -q^2-2 q^{-2} + q^{-6} +2 q^{-8} +6 q^{-10} +2 q^{-12} +4 q^{-14} -2 q^{-16} -3 q^{-18} -3 q^{-20} -3 q^{-22} + q^{-24} + q^{-28}
The G2 invariant Data:K11n78/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_62, K11n76,}

Same Jones Polynomial (up to mirroring, q\leftrightarrow q^{-1}): {K11n76,}

Vassiliev invariants

V2 and V3: (5, 7)
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
20 56 200 \frac{886}{3} \frac{98}{3} 1120 \frac{4496}{3} \frac{704}{3} 152 \frac{4000}{3} 1568 \frac{17720}{3} \frac{1960}{3} \frac{50767}{6} \frac{2090}{3} \frac{21446}{9} \frac{853}{18} \frac{1423}{6}

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 K11n78. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
  \  
j \
-3-2-10123456χ
17         11
15        3 -3
13       21 1
11      53  -2
9     32   1
7    35    2
5   43     1
3  14      3
1 13       -2
-1 1        1
-31         -1
Integral Khovanov Homology

(db, data source)

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

Back to the top.

K11n77.gif

K11n77

K11n79.gif

K11n79