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

K11a120 K11a122

Knotscape

This page is passe. Go here instead!    The Knot K11a121 Visit K11a121's page at Knotilus! Acknowledgement

DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X14,5,15,6 X18,7,19,8 X12,10,13,9 X2,11,3,12 X22,13,1,14 X8,15,9,16 X20,18,21,17 X6,19,7,20 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 18 12 2 22 8 20 6 16

Alexander Polynomial:

t-3 - 9t-2 + 29t-1 - 41 + 29t - 9t2 + t3

Conway Polynomial:

1 + 2z2 - 3z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{119, -2}

Jones Polynomial:

- q-8 + 3q-7 - 7q-6 + 12q-5 - 16q-4 + 19q-3 - 19q-2 + 17q-1 - 12 + 8q - 4q2 + q3

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

{K11a66, ...}

A2 (sl(3)) Invariant:

- q-26 - q-24 + 2q-22 - q-20 - q-18 + 4q-16 - 3q-14 + q-12 + q-10 - 2q-8 + 3q-6 - 3q-4 + 4q-2 + 1 - 2q2 + 3q4 - 2q6 - q8 + q10

HOMFLY-PT Polynomial:

a-2z2 - 2z2 - 2z4 + 2a2 + 4a2z2 + 2a2z4 + a2z6 - 2a4 - 4a4z2 - 3a4z4 + 2a6 + 3a6z2 - a8

Kauffman Polynomial:

a-2z2 - 2a-2z4 + a-2z6 + 6a-1z3 - 10a-1z5 + 4a-1z7 + 8z4 - 14z6 + 6z8 + 7az3 - 12az5 - az7 + 4az9 - 2a2 + 2a2z2 + 11a2z4 - 25a2z6 + 11a2z8 + a2z10 - a3z + 9a3z3 - 13a3z5 - 3a3z7 + 7a3z9 - 2a4 + 7a4z2 - 15a4z6 + 10a4z8 + a4z10 - 3a5z + 14a5z3 - 19a5z5 + 7a5z7 + 3a5z9 - 2a6 + 7a6z2 - 6a6z4 - 2a6z6 + 5a6z8 - a7z + 4a7z3 - 7a7z5 + 5a7z7 - a8 + 3a8z2 - 5a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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

{2, -4}

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=-2 is the signature of 11121. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 7                       1 j = 5                     3   j = 3                   5 1   j = 1                 7 3     j = -1               10 5       j = -3             10 8         j = -5           9 9           j = -7         7 10             j = -9       5 9               j = -11     2 7                 j = -13   1 5                   j = -15   2                     j = -17 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, Alternating, 121]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 121]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 5, 15, 6], X[18, 7, 19, 8], > X[12, 10, 13, 9], X[2, 11, 3, 12], X[22, 13, 1, 14], X[8, 15, 9, 16], > X[20, 18, 21, 17], X[6, 19, 7, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 121]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -10, 4, -8, 5, -2, 6, -5, 7, -3, 8, -11, 9, -4, 10, > -9, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 121]]
Out[5]=	DTCode[4, 10, 14, 18, 12, 2, 22, 8, 20, 6, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 121]][t]
Out[6]=	-3 9 29 2 3 -41 + t - -- + -- + 29 t - 9 t + t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 121]][z]
Out[7]=	2 4 6 1 + 2 z - 3 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 121]}
In[9]:=	{KnotDet[Knot[11, Alternating, 121]], KnotSignature[Knot[11, Alternating, 121]]}
Out[9]=	{119, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 121]][q]
Out[10]=	-8 3 7 12 16 19 19 17 2 3 -12 - q + -- - -- + -- - -- + -- - -- + -- + 8 q - 4 q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 66], Knot[11, Alternating, 121]}
In[12]:=	A2Invariant[Knot[11, Alternating, 121]][q]
Out[12]=	-26 -24 2 -20 -18 4 3 -12 -10 2 3 3 1 - q - q + --- - q - q + --- - --- + q + q - -- + -- - -- + 22 16 14 8 6 4 q q q q q q 4 2 4 6 8 10 > -- - 2 q + 3 q - 2 q - q + q 2 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 121]][a, z]
Out[13]=	2 2 4 6 8 2 z 2 2 4 2 6 2 4 2 a - 2 a + 2 a - a - 2 z + -- + 4 a z - 4 a z + 3 a z - 2 z + 2 a 2 4 4 4 2 6 > 2 a z - 3 a z + a z
In[14]:=	Kauffman[Knot[11, Alternating, 121]][a, z]
Out[14]=	2 2 4 6 8 3 5 7 9 z 2 2 -2 a - 2 a - 2 a - a - a z - 3 a z - a z + a z + -- + 2 a z + 2 a 3 4 2 6 2 8 2 6 z 3 3 3 5 3 > 7 a z + 7 a z + 3 a z + ---- + 7 a z + 9 a z + 14 a z + a 4 5 7 3 9 3 4 2 z 2 4 6 4 8 4 10 z > 4 a z - 2 a z + 8 z - ---- + 11 a z - 6 a z - 5 a z - ----- - 2 a a 6 5 3 5 5 5 7 5 9 5 6 z 2 6 > 12 a z - 13 a z - 19 a z - 7 a z + a z - 14 z + -- - 25 a z - 2 a 7 4 6 6 6 8 6 4 z 7 3 7 5 7 7 7 > 15 a z - 2 a z + 3 a z + ---- - a z - 3 a z + 7 a z + 5 a z + a 8 2 8 4 8 6 8 9 3 9 5 9 > 6 z + 11 a z + 10 a z + 5 a z + 4 a z + 7 a z + 3 a z + 2 10 4 10 > a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 121]], Vassiliev[3][Knot[11, Alternating, 121]]}
Out[15]=	{2, -4}
In[16]:=	Kh[Knot[11, Alternating, 121]][q, t]
Out[16]=	8 10 1 2 1 5 2 7 5 9 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 7 10 9 9 10 5 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 7 q t + 3 q t + 5 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a121

K11a120 K11a122