© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n13 K11n15

Knotscape

This page is passe. Go here instead!    The Knot K11n14 Visit K11n14's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X7,16,8,17 X2,9,3,10 X18,11,19,12 X20,13,21,14 X15,6,16,7 X22,17,1,18 X12,19,13,20 X14,21,15,22

Gauss Code:

{1, -5, 2, -1, 3, 8, -4, -2, 5, -3, 6, -10, 7, -11, -8, 4, 9, -6, 10, -7, 11, -9}

DT (Dowker-Thistlethwaite) Code:

4 8 10 -16 2 18 20 -6 22 12 14

Alexander Polynomial:

- t-3 + 6t-2 - 10t-1 + 11 - 10t + 6t2 - t3

Conway Polynomial:

1 + 5z2 - z6

Other knots with the same Alexander/Conway Polynomial:

{K11n121, ...}

Determinant and Signature:

{45, -4}

Jones Polynomial:

- q-11 + 2q-10 - 4q-9 + 6q-8 - 7q-7 + 8q-6 - 7q-5 + 5q-4 - 3q-3 + 2q-2

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-34 - q-32 - q-28 + 2q-26 + q-24 + q-20 - 2q-18 + q-16 - q-14 + 2q-10 + 2q-6

HOMFLY-PT Polynomial:

3a4 + 6a4z2 + 2a4z4 - 4a6 - 6a6z2 - 4a6z4 - a6z6 + 4a8 + 6a8z2 + 2a8z4 - 2a10 - a10z2

Kauffman Polynomial:

3a4 - 8a4z2 + 3a4z4 - a5z3 - a5z5 + a5z7 + 4a6 - 15a6z2 + 17a6z4 - 8a6z6 + 2a6z8 + a7z + 3a7z5 - 2a7z7 + a7z9 + 4a8 - 17a8z2 + 27a8z4 - 15a8z6 + 4a8z8 - a9z + 3a9z3 - a9z7 + a9z9 + 2a10 - 8a10z2 + 8a10z4 - 5a10z6 + 2a10z8 - a11z3 - 3a11z5 + 2a11z7 + 2a12z2 - 5a12z4 + 2a12z6 + 2a13z - 3a13z3 + a13z5

V2 and V3, the type 2 and 3 Vassiliev invariants:

{5, -13}

Khovanov Homology: (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 1114. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -3                   2 j = -5                 2 1 j = -7               3 1   j = -9             4 2     j = -11           4 3       j = -13         3 4         j = -15       3 4           j = -17     1 3             j = -19   1 3               j = -21   1                 j = -23 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 14]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 14]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[7, 16, 8, 17], > X[2, 9, 3, 10], X[18, 11, 19, 12], X[20, 13, 21, 14], X[15, 6, 16, 7], > X[22, 17, 1, 18], X[12, 19, 13, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 14]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, 8, -4, -2, 5, -3, 6, -10, 7, -11, -8, 4, 9, -6, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 14]]
Out[5]=	DTCode[4, 8, 10, -16, 2, 18, 20, -6, 22, 12, 14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 14]][t]
Out[6]=	-3 6 10 2 3 11 - t + -- - -- - 10 t + 6 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 14]][z]
Out[7]=	2 6 1 + 5 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 14], Knot[11, NonAlternating, 121]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 14]], KnotSignature[Knot[11, NonAlternating, 14]]}
Out[9]=	{45, -4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 14]][q]
Out[10]=	-11 2 4 6 7 8 7 5 3 2 -q + --- - -- + -- - -- + -- - -- + -- - -- + -- 10 9 8 7 6 5 4 3 2 q q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 14]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 14]][q]
Out[12]=	-34 -32 -28 2 -24 -20 2 -16 -14 2 2 -q - q - q + --- + q + q - --- + q - q + --- + -- 26 18 10 6 q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 14]][a, z]
Out[13]=	4 6 8 10 4 2 6 2 8 2 10 2 4 4 3 a - 4 a + 4 a - 2 a + 6 a z - 6 a z + 6 a z - a z + 2 a z - 6 4 8 4 6 6 > 4 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 14]][a, z]
Out[14]=	4 6 8 10 7 9 13 4 2 6 2 3 a + 4 a + 4 a + 2 a + a z - a z + 2 a z - 8 a z - 15 a z - 8 2 10 2 12 2 5 3 9 3 11 3 13 3 > 17 a z - 8 a z + 2 a z - a z + 3 a z - a z - 3 a z + 4 4 6 4 8 4 10 4 12 4 5 5 7 5 > 3 a z + 17 a z + 27 a z + 8 a z - 5 a z - a z + 3 a z - 11 5 13 5 6 6 8 6 10 6 12 6 5 7 > 3 a z + a z - 8 a z - 15 a z - 5 a z + 2 a z + a z - 7 7 9 7 11 7 6 8 8 8 10 8 7 9 9 9 > 2 a z - a z + 2 a z + 2 a z + 4 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 14]], Vassiliev[3][Knot[11, NonAlternating, 14]]}
Out[15]=	{5, -13}
In[16]:=	Kh[Knot[11, NonAlternating, 14]][q, t]
Out[16]=	-5 2 1 1 1 3 1 3 3 q + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q t q t q t q t q t q t q t 4 3 4 4 3 4 2 3 1 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 2 > ---- 5 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n14

K11n13 K11n15