From Knot Atlas
Jump to: navigation, search






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

Visit K11n179 at Knotilus!

Knot presentations

Planar diagram presentation X6271 X3,10,4,11 X5,20,6,21 X14,8,15,7 X9,2,10,3 X11,19,12,18 X8,14,9,13 X22,16,1,15 X17,13,18,12 X19,4,20,5 X16,22,17,21
Gauss code 1, 5, -2, 10, -3, -1, 4, -7, -5, 2, -6, 9, 7, -4, 8, -11, -9, 6, -10, 3, 11, -8
Dowker-Thistlethwaite code 6 -10 -20 14 -2 -18 8 22 -12 -4 16
A Braid Representative
A Morse Link Presentation K11n179 ML.gif

Three dimensional invariants

Symmetry type Reversible
Unknotting number \{1,2\}
3-genus 3
Bridge index 3
Super bridge index Missing
Nakanishi index Missing
Maximal Thurston-Bennequin number Data:K11n179/ThurstonBennequinNumber
Hyperbolic Volume 14.7989
A-Polynomial See Data:K11n179/A-polynomial

[edit Notes for K11n179's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing
Topological 4 genus Missing
Concordance genus 3
Rasmussen s-Invariant 0

[edit Notes for K11n179's four dimensional invariants]

Polynomial invariants

Alexander polynomial -t^3+7 t^2-18 t+25-18 t^{-1} +7 t^{-2} - t^{-3}
Conway polynomial -z^6+z^4+z^2+1
2nd Alexander ideal (db, data sources) \{1\}
Determinant and Signature { 77, 0 }
Jones polynomial -2 q^5+5 q^4-8 q^3+12 q^2-13 q+13-11 q^{-1} +8 q^{-2} -4 q^{-3} + q^{-4}
HOMFLY-PT polynomial (db, data sources) -z^6+a^2 z^4+3 z^4 a^{-2} -3 z^4+a^2 z^2+7 z^2 a^{-2} -2 z^2 a^{-4} -5 z^2+a^2+5 a^{-2} -2 a^{-4} -3
Kauffman polynomial (db, data sources) z^9 a^{-1} +z^9 a^{-3} +6 z^8 a^{-2} +z^8 a^{-4} +5 z^8+9 a z^7+12 z^7 a^{-1} +3 z^7 a^{-3} +8 a^2 z^6-4 z^6 a^{-2} +2 z^6 a^{-4} +2 z^6+4 a^3 z^5-12 a z^5-26 z^5 a^{-1} -7 z^5 a^{-3} +3 z^5 a^{-5} +a^4 z^4-9 a^2 z^4-15 z^4 a^{-2} -9 z^4 a^{-4} -16 z^4-2 a^3 z^3+5 a z^3+13 z^3 a^{-1} -6 z^3 a^{-5} +3 a^2 z^2+17 z^2 a^{-2} +8 z^2 a^{-4} +12 z^2-a z-z a^{-1} +2 z a^{-3} +2 z a^{-5} -a^2-5 a^{-2} -2 a^{-4} -3
The A2 invariant Data:K11n179/QuantumInvariant/A2/1,0
The G2 invariant Data:K11n179/QuantumInvariant/G2/1,0

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {10_71, K11n156,}

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{62}{3} -\frac{14}{3} 64 \frac{352}{3} -\frac{128}{3} 80 \frac{32}{3} 128 \frac{248}{3} -\frac{56}{3} \frac{11311}{30} -\frac{634}{5} \frac{8942}{45} \frac{1361}{18} \frac{751}{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 K11n179. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.   
\ r
j \
11         2-2
9        3 3
7       52 -3
5      73  4
3     65   -1
1    77    0
-1   57     2
-3  36      -3
-5 15       4
-7 3        -3
-91         1
Integral Khovanov Homology

(db, data source)

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

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.