K11a14

K11a13

K11a15

(Knotscape image)	See the full Hoste-Thistlethwaite Table of 11 Crossing Knots. Visit K11a14 at Knotilus!

Knot presentations

Planar diagram presentation	X4251 X8493 X12,5,13,6 X2837 X14,9,15,10 X18,11,19,12 X6,13,7,14 X20,16,21,15 X10,17,11,18 X22,20,1,19 X16,22,17,21
Gauss code	1, -4, 2, -1, 3, -7, 4, -2, 5, -9, 6, -3, 7, -5, 8, -11, 9, -6, 10, -8, 11, -10
Dowker-Thistlethwaite code	4 8 12 2 14 18 6 20 10 22 16

A Braid Representative
A Morse Link Presentation

Three dimensional invariants

Symmetry type Reversible Unknotting number [math]\displaystyle{ \{1,2,3\} }[/math] 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a14/ThurstonBennequinNumber Hyperbolic Volume 16.4066 A-Polynomial See Data:K11a14/A-polynomial

[edit Notes for K11a14's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus [math]\displaystyle{ 4 }[/math] Rasmussen s-Invariant 0

[edit Notes for K11a14's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ t^4-5 t^3+15 t^2-28 t+35-28 t^{-1} +15 t^{-2} -5 t^{-3} + t^{-4} }[/math]
Conway polynomial	[math]\displaystyle{ z^8+3 z^6+5 z^4+3 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 133, 0 }
Jones polynomial	[math]\displaystyle{ q^6-4 q^5+8 q^4-14 q^3+19 q^2-21 q+22-18 q^{-1} +14 q^{-2} -8 q^{-3} +3 q^{-4} - q^{-5} }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +16 z^4-7 a^2 z^2-12 z^2 a^{-2} +2 z^2 a^{-4} +20 z^2-4 a^2-6 a^{-2} + a^{-4} +10 }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^{10} a^{-2} +z^{10}+5 a z^9+9 z^9 a^{-1} +4 z^9 a^{-3} +8 a^2 z^8+17 z^8 a^{-2} +6 z^8 a^{-4} +19 z^8+6 a^3 z^7+4 a z^7-z^7 a^{-1} +5 z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-14 a^2 z^6-46 z^6 a^{-2} -12 z^6 a^{-4} +z^6 a^{-6} -50 z^6+a^5 z^5-9 a^3 z^5-24 a z^5-36 z^5 a^{-1} -32 z^5 a^{-3} -10 z^5 a^{-5} -4 a^4 z^4+17 a^2 z^4+41 z^4 a^{-2} +6 z^4 a^{-4} -2 z^4 a^{-6} +54 z^4-2 a^5 z^3+7 a^3 z^3+29 a z^3+43 z^3 a^{-1} +31 z^3 a^{-3} +8 z^3 a^{-5} +a^4 z^2-13 a^2 z^2-21 z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} -32 z^2+a^5 z-3 a^3 z-12 a z-16 z a^{-1} -10 z a^{-3} -2 z a^{-5} +4 a^2+6 a^{-2} + a^{-4} +10 }[/math]
The A2 invariant	[math]\displaystyle{ -q^{14}+q^{12}-4 q^{10}+q^8+q^6-2 q^4+7 q^2-1+5 q^{-2} -2 q^{-6} +2 q^{-8} -5 q^{-10} + q^{-12} - q^{-16} + q^{-18} }[/math]
The G2 invariant	[math]\displaystyle{ q^{80}-2 q^{78}+5 q^{76}-8 q^{74}+10 q^{72}-10 q^{70}+4 q^{68}+10 q^{66}-29 q^{64}+52 q^{62}-74 q^{60}+78 q^{58}-61 q^{56}+7 q^{54}+89 q^{52}-198 q^{50}+290 q^{48}-316 q^{46}+225 q^{44}-31 q^{42}-245 q^{40}+508 q^{38}-649 q^{36}+586 q^{34}-312 q^{32}-108 q^{30}+511 q^{28}-748 q^{26}+712 q^{24}-413 q^{22}-30 q^{20}+427 q^{18}-607 q^{16}+494 q^{14}-126 q^{12}-308 q^{10}+628 q^8-659 q^6+373 q^4+140 q^2-660+1001 q^{-2} -986 q^{-4} +624 q^{-6} -15 q^{-8} -611 q^{-10} +1037 q^{-12} -1105 q^{-14} +803 q^{-16} -249 q^{-18} -340 q^{-20} +734 q^{-22} -798 q^{-24} +534 q^{-26} -76 q^{-28} -370 q^{-30} +589 q^{-32} -511 q^{-34} +163 q^{-36} +282 q^{-38} -631 q^{-40} +730 q^{-42} -540 q^{-44} +132 q^{-46} +313 q^{-48} -642 q^{-50} +740 q^{-52} -591 q^{-54} +284 q^{-56} +60 q^{-58} -328 q^{-60} +447 q^{-62} -411 q^{-64} +273 q^{-66} -95 q^{-68} -52 q^{-70} +134 q^{-72} -153 q^{-74} +123 q^{-76} -69 q^{-78} +24 q^{-80} +9 q^{-82} -23 q^{-84} +21 q^{-86} -16 q^{-88} +8 q^{-90} -3 q^{-92} + q^{-94} }[/math]

Further Quantum Invariants

K11a14 Quantum Invariants.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a14"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

[math]\displaystyle{ t^4-5 t^3+15 t^2-28 t+35-28 t^{-1} +15 t^{-2} -5 t^{-3} + t^{-4} }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ z^8+3 z^6+5 z^4+3 z^2+1 }[/math]

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 133, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

[math]\displaystyle{ q^6-4 q^5+8 q^4-14 q^3+19 q^2-21 q+22-18 q^{-1} +14 q^{-2} -8 q^{-3} +3 q^{-4} - q^{-5} }[/math]

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

[math]\displaystyle{ z^8-a^2 z^6-2 z^6 a^{-2} +6 z^6-4 a^2 z^4-8 z^4 a^{-2} +z^4 a^{-4} +16 z^4-7 a^2 z^2-12 z^2 a^{-2} +2 z^2 a^{-4} +20 z^2-4 a^2-6 a^{-2} + a^{-4} +10 }[/math]

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

[math]\displaystyle{ z^{10} a^{-2} +z^{10}+5 a z^9+9 z^9 a^{-1} +4 z^9 a^{-3} +8 a^2 z^8+17 z^8 a^{-2} +6 z^8 a^{-4} +19 z^8+6 a^3 z^7+4 a z^7-z^7 a^{-1} +5 z^7 a^{-3} +4 z^7 a^{-5} +3 a^4 z^6-14 a^2 z^6-46 z^6 a^{-2} -12 z^6 a^{-4} +z^6 a^{-6} -50 z^6+a^5 z^5-9 a^3 z^5-24 a z^5-36 z^5 a^{-1} -32 z^5 a^{-3} -10 z^5 a^{-5} -4 a^4 z^4+17 a^2 z^4+41 z^4 a^{-2} +6 z^4 a^{-4} -2 z^4 a^{-6} +54 z^4-2 a^5 z^3+7 a^3 z^3+29 a z^3+43 z^3 a^{-1} +31 z^3 a^{-3} +8 z^3 a^{-5} +a^4 z^2-13 a^2 z^2-21 z^2 a^{-2} -2 z^2 a^{-4} +z^2 a^{-6} -32 z^2+a^5 z-3 a^3 z-12 a z-16 z a^{-1} -10 z a^{-3} -2 z a^{-5} +4 a^2+6 a^{-2} + a^{-4} +10 }[/math]

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, [math]\displaystyle{ q\leftrightarrow q^{-1} }[/math]): {}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["K11a14"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ [math]\displaystyle{ t^4-5 t^3+15 t^2-28 t+35-28 t^{-1} +15 t^{-2} -5 t^{-3} + t^{-4} }[/math], [math]\displaystyle{ q^6-4 q^5+8 q^4-14 q^3+19 q^2-21 q+22-18 q^{-1} +14 q^{-2} -8 q^{-3} +3 q^{-4} - q^{-5} }[/math] }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{}

Vassiliev invariants

V2 and V3:

(3, -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 [math]\displaystyle{ 12 }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 72 }[/math] [math]\displaystyle{ 62 }[/math] [math]\displaystyle{ -6 }[/math] [math]\displaystyle{ -96 }[/math] [math]\displaystyle{ -\frac{368}{3} }[/math] [math]\displaystyle{ -\frac{32}{3} }[/math] [math]\displaystyle{ -8 }[/math] [math]\displaystyle{ 288 }[/math] [math]\displaystyle{ 32 }[/math] [math]\displaystyle{ 744 }[/math] [math]\displaystyle{ -72 }[/math] [math]\displaystyle{ \frac{7311}{10} }[/math] [math]\displaystyle{ \frac{2974}{15} }[/math] [math]\displaystyle{ -\frac{698}{15} }[/math] [math]\displaystyle{ \frac{145}{6} }[/math] [math]\displaystyle{ -\frac{689}{10} }[/math]

V_2,1 through V_6,9 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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s-1 }[/math], where [math]\displaystyle{ s= }[/math]0 is the signature of K11a14. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 13                       1 1 11                     3   -3 9                   5 1   4 7                 9 3     -6 5               10 5       5 3             11 9         -2 1           11 10           1 -1         8 12             4 -3       6 10               -4 -5     2 8                 6 -7   1 6                   -5 -9   2                     2 -11 1                       -1

Integral Khovanov Homology (db, data source)

[math]\displaystyle{ \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z} }[/math] [math]\displaystyle{ i=-1 }[/math] [math]\displaystyle{ i=1 }[/math] [math]\displaystyle{ r=-5 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-4 }[/math] [math]\displaystyle{ {\mathbb Z}^{2}\oplus{\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math] [math]\displaystyle{ r=-3 }[/math] [math]\displaystyle{ {\mathbb Z}^{6}\oplus{\mathbb Z}_2^{2} }[/math] [math]\displaystyle{ {\mathbb Z}^{2} }[/math] [math]\displaystyle{ r=-2 }[/math] [math]\displaystyle{ {\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6} }[/math] [math]\displaystyle{ {\mathbb Z}^{6} }[/math] [math]\displaystyle{ r=-1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{8} }[/math] [math]\displaystyle{ {\mathbb Z}^{8} }[/math] [math]\displaystyle{ r=0 }[/math] [math]\displaystyle{ {\mathbb Z}^{12}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=1 }[/math] [math]\displaystyle{ {\mathbb Z}^{10}\oplus{\mathbb Z}_2^{11} }[/math] [math]\displaystyle{ {\mathbb Z}^{11} }[/math] [math]\displaystyle{ r=2 }[/math] [math]\displaystyle{ {\mathbb Z}^{9}\oplus{\mathbb Z}_2^{10} }[/math] [math]\displaystyle{ {\mathbb Z}^{10} }[/math] [math]\displaystyle{ r=3 }[/math] [math]\displaystyle{ {\mathbb Z}^{5}\oplus{\mathbb Z}_2^{9} }[/math] [math]\displaystyle{ {\mathbb Z}^{9} }[/math] [math]\displaystyle{ r=4 }[/math] [math]\displaystyle{ {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5} }[/math] [math]\displaystyle{ {\mathbb Z}^{5} }[/math] [math]\displaystyle{ r=5 }[/math] [math]\displaystyle{ {\mathbb Z}\oplus{\mathbb Z}_2^{3} }[/math] [math]\displaystyle{ {\mathbb Z}^{3} }[/math] [math]\displaystyle{ r=6 }[/math] [math]\displaystyle{ {\mathbb Z}_2 }[/math] [math]\displaystyle{ {\mathbb Z} }[/math]

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.

K11a13 K11a15