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

K11a3 K11a5

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X14,8,15,7 X2,9,3,10 X18,11,19,12 X6,14,7,13 X22,15,1,16 X20,17,21,18 X12,19,13,20 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 10 14 2 18 6 22 20 12 16

Alexander Polynomial:

- 2t-3 + 10t-2 - 22t-1 + 29 - 22t + 10t2 - 2t3

Conway Polynomial:

1 - 2z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{K11a110, ...}

Determinant and Signature:

{97, 0}

Jones Polynomial:

- q-7 + 3q-6 - 6q-5 + 10q-4 - 13q-3 + 15q-2 - 15q-1 + 14 - 10q + 6q2 - 3q3 + q4

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

{...}

A2 (sl(3)) Invariant:

- q-22 + q-18 - 2q-16 + 2q-14 + q-12 - q-10 + 3q-8 - 2q-6 + q-4 - 1 + 3q2 - 3q4 + q6 + q8 - q10 + q12

HOMFLY-PT Polynomial:

a-2 + 2a-2z2 + a-2z4 - 1 - 4z2 - 3z4 - z6 - a2z2 - 2a2z4 - a2z6 + 2a4 + 4a4z2 + 2a4z4 - a6 - a6z2

Kauffman Polynomial:

- a-4z2 + a-4z4 + a-3z - 3a-3z3 + 3a-3z5 - a-2 + 3a-2z2 - 5a-2z4 + 5a-2z6 + a-1z - 5a-1z5 + 6a-1z7 - 1 + 5z2 - 2z4 - 6z6 + 6z8 - az + 7az3 - 10az5 - az7 + 4az9 - 8a2z2 + 28a2z4 - 33a2z6 + 10a2z8 + a2z10 - 3a3z + 10a3z3 - a3z5 - 15a3z7 + 7a3z9 + 2a4 - 16a4z2 + 39a4z4 - 34a4z6 + 7a4z8 + a4z10 - 4a5z + 11a5z3 - 3a5z5 - 7a5z7 + 3a5z9 + a6 - 7a6z2 + 15a6z4 - 12a6z6 + 3a6z8 - 2a7z + 5a7z3 - 4a7z5 + a7z7

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

{0, -2}

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 114. 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 = 9                       1 j = 7                     2   j = 5                   4 1   j = 3                 6 2     j = 1               8 4       j = -1             8 7         j = -3           7 7           j = -5         6 8             j = -7       4 7               j = -9     2 6                 j = -11   1 4                   j = -13   2                     j = -15 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, 4]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 4]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[14, 8, 15, 7], > X[2, 9, 3, 10], X[18, 11, 19, 12], X[6, 14, 7, 13], X[22, 15, 1, 16], > X[20, 17, 21, 18], X[12, 19, 13, 20], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 4]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -7, 4, -2, 5, -3, 6, -10, 7, -4, 8, -11, 9, -6, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 4]]
Out[5]=	DTCode[4, 8, 10, 14, 2, 18, 6, 22, 20, 12, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 4]][t]
Out[6]=	2 10 22 2 3 29 - -- + -- - -- - 22 t + 10 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 4]][z]
Out[7]=	4 6 1 - 2 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 4], Knot[11, Alternating, 110]}
In[9]:=	{KnotDet[Knot[11, Alternating, 4]], KnotSignature[Knot[11, Alternating, 4]]}
Out[9]=	{97, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 4]][q]
Out[10]=	-7 3 6 10 13 15 15 2 3 4 14 - q + -- - -- + -- - -- + -- - -- - 10 q + 6 q - 3 q + q 6 5 4 3 2 q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 4]}
In[12]:=	A2Invariant[Knot[11, Alternating, 4]][q]
Out[12]=	-22 -18 2 2 -12 -10 3 2 -4 2 4 6 -1 - q + q - --- + --- + q - q + -- - -- + q + 3 q - 3 q + q + 16 14 8 6 q q q q 8 10 12 > q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 4]][a, z]
Out[13]=	2 4 -2 4 6 2 2 z 2 2 4 2 6 2 4 z -1 + a + 2 a - a - 4 z + ---- - a z + 4 a z - a z - 3 z + -- - 2 2 a a 2 4 4 4 6 2 6 > 2 a z + 2 a z - z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 4]][a, z]
Out[14]=	2 -2 4 6 z z 3 5 7 2 z -1 - a + 2 a + a + -- + - - a z - 3 a z - 4 a z - 2 a z + 5 z - -- + 3 a 4 a a 2 3 3 z 2 2 4 2 6 2 3 z 3 3 3 5 3 > ---- - 8 a z - 16 a z - 7 a z - ---- + 7 a z + 10 a z + 11 a z + 2 3 a a 4 4 5 5 7 3 4 z 5 z 2 4 4 4 6 4 3 z 5 z > 5 a z - 2 z + -- - ---- + 28 a z + 39 a z + 15 a z + ---- - ---- - 4 2 3 a a a a 6 5 3 5 5 5 7 5 6 5 z 2 6 4 6 > 10 a z - a z - 3 a z - 4 a z - 6 z + ---- - 33 a z - 34 a z - 2 a 7 6 6 6 z 7 3 7 5 7 7 7 8 2 8 > 12 a z + ---- - a z - 15 a z - 7 a z + a z + 6 z + 10 a z + a 4 8 6 8 9 3 9 5 9 2 10 4 10 > 7 a z + 3 a z + 4 a z + 7 a z + 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 4]], Vassiliev[3][Knot[11, Alternating, 4]]}
Out[15]=	{0, -2}
In[16]:=	Kh[Knot[11, Alternating, 4]][q, t]
Out[16]=	7 1 2 1 4 2 6 4 7 - + 8 q + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ----- + q 15 7 13 6 11 6 11 5 9 5 9 4 7 4 7 3 q t q t q t q t q t q t q t q t 6 8 7 7 8 3 3 2 5 2 > ----- + ----- + ----- + ---- + --- + 4 q t + 6 q t + 2 q t + 4 q t + 5 3 5 2 3 2 3 q t q t q t q t q t 5 3 7 3 9 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a4

K11a3 K11a5