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

K11n97 K11n99

Knotscape

This page is passe. Go here instead!    The Knot K11n98 Visit K11n98's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,14,6,15 X7,12,8,13 X18,9,19,10 X2,11,3,12 X13,6,14,7 X20,16,21,15 X22,18,1,17 X8,19,9,20 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 7t-2 - 16t-1 + 21 - 16t + 7t2 - t3

Conway Polynomial:

1 + 3z2 + z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{1078, K11n105, ...}

Determinant and Signature:

{69, 0}

Jones Polynomial:

- 2q-5 + 4q-4 - 7q-3 + 11q-2 - 11q-1 + 12 - 10q + 7q2 - 4q3 + q4

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

{...}

A2 (sl(3)) Invariant:

- 2q-16 - q-14 - 2q-10 + 4q-8 + 2q-6 + 2q-4 + 3q-2 - 2 + 2q2 - 3q4 + q8 - 2q10 + q12

HOMFLY-PT Polynomial:

a-2z2 + a-2z4 - 2 - 4z2 - 3z4 - z6 + 6a2 + 8a2z2 + 3a2z4 - 3a4 - 2a4z2

Kauffman Polynomial:

a-4z4 - 3a-3z3 + 4a-3z5 + 2a-2z2 - 8a-2z4 + 7a-2z6 + 2a-1z - a-1z3 - 7a-1z5 + 7a-1z7 - 2 + 6z2 - 9z4 + z6 + 4z8 + 6az - 3az3 - 11az5 + 7az7 + az9 - 6a2 + 9a2z2 - 4a2z4 - 6a2z6 + 5a2z8 + 9a3z - 13a3z3 + 3a3z5 + a3z9 - 3a4 + 5a4z2 - 4a4z4 + a4z8 + 5a5z - 8a5z3 + 3a5z5

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

{3, -3}

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=0 is the signature of 1198. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                   1 j = 7                 3   j = 5               4 1   j = 3             6 3     j = 1           6 4       j = -1         6 7         j = -3       5 5           j = -5     2 6             j = -7   2 5               j = -9   2                 j = -11 2

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, 98]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 98]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 14, 6, 15], X[7, 12, 8, 13], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[13, 6, 14, 7], X[20, 16, 21, 15], > X[22, 18, 1, 17], X[8, 19, 9, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 98]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 7, -4, -10, 5, -2, 6, 4, -7, 3, 8, -11, 9, -5, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 98]]
Out[5]=	DTCode[4, 10, -14, -12, 18, 2, -6, 20, 22, 8, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 98]][t]
Out[6]=	-3 7 16 2 3 21 - t + -- - -- - 16 t + 7 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 98]][z]
Out[7]=	2 4 6 1 + 3 z + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 78], Knot[11, NonAlternating, 98], Knot[11, NonAlternating, 105]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 98]], KnotSignature[Knot[11, NonAlternating, 98]]}
Out[9]=	{69, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 98]][q]
Out[10]=	2 4 7 11 11 2 3 4 12 - -- + -- - -- + -- - -- - 10 q + 7 q - 4 q + q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 98]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 98]][q]
Out[12]=	2 -14 2 4 2 2 3 2 4 8 10 12 -2 - --- - q - --- + -- + -- + -- + -- + 2 q - 3 q + q - 2 q + q 16 10 8 6 4 2 q q q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 98]][a, z]
Out[13]=	2 4 2 4 2 z 2 2 4 2 4 z 2 4 6 -2 + 6 a - 3 a - 4 z + -- + 8 a z - 2 a z - 3 z + -- + 3 a z - z 2 2 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 98]][a, z]
Out[14]=	2 2 4 2 z 3 5 2 2 z 2 2 -2 - 6 a - 3 a + --- + 6 a z + 9 a z + 5 a z + 6 z + ---- + 9 a z + a 2 a 3 3 4 4 4 2 3 z z 3 3 3 5 3 4 z 8 z > 5 a z - ---- - -- - 3 a z - 13 a z - 8 a z - 9 z + -- - ---- - 3 a 4 2 a a a 5 5 6 2 4 4 4 4 z 7 z 5 3 5 5 5 6 7 z > 4 a z - 4 a z + ---- - ---- - 11 a z + 3 a z + 3 a z + z + ---- - 3 a 2 a a 7 2 6 7 z 7 8 2 8 4 8 9 3 9 > 6 a z + ---- + 7 a z + 4 z + 5 a z + a z + a z + a z a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 98]], Vassiliev[3][Knot[11, NonAlternating, 98]]}
Out[15]=	{3, -3}
In[16]:=	Kh[Knot[11, NonAlternating, 98]][q, t]
Out[16]=	7 2 2 2 5 2 6 5 5 6 - + 6 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 9 4 > 4 q t + 6 q t + 3 q t + 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n98

K11n97 K11n99