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

K11a47 K11a49

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 10 2 16 20 6 22 12 18

Alexander Polynomial:

- 2t-3 + 12t-2 - 26t-1 + 33 - 26t + 12t2 - 2t3

Conway Polynomial:

1 + 4z2 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{113, -4}

Jones Polynomial:

- q-11 + 3q-10 - 7q-9 + 12q-8 - 16q-7 + 18q-6 - 18q-5 + 16q-4 - 11q-3 + 7q-2 - 3q-1 + 1

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

{...}

A2 (sl(3)) Invariant:

- q-34 + q-30 - 3q-28 + 2q-26 + q-24 - 2q-22 + 3q-20 - 3q-18 + 2q-16 - q-12 + 4q-10 - 3q-8 + 2q-6 + q-4 - q-2 + 1

HOMFLY-PT Polynomial:

a2 + 2a2z2 + a2z4 - a4z2 - 2a4z4 - a4z6 + a6z2 - a6z4 - a6z6 + a8 + 3a8z2 + 2a8z4 - a10 - a10z2

Kauffman Polynomial:

- a2 + 3a2z2 - 3a2z4 + a2z6 - a3z + 6a3z3 - 8a3z5 + 3a3z7 + 2a4z2 - 7a4z6 + 4a4z8 - 3a5z + 11a5z3 - 14a5z5 + a5z7 + 3a5z9 - 4a6z2 + 16a6z4 - 23a6z6 + 9a6z8 + a6z10 - 2a7z + 7a7z3 - 5a7z5 - 7a7z7 + 7a7z9 + a8 - 10a8z2 + 26a8z4 - 27a8z6 + 11a8z8 + a8z10 - 2a9z + 7a9z3 - 7a9z5 + 4a9z9 + a10 - 5a10z2 + 8a10z4 - 9a10z6 + 6a10z8 - a11z + 3a11z3 - 7a11z5 + 5a11z7 + 2a12z2 - 5a12z4 + 3a12z6 + a13z - 2a13z3 + a13z5

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

{4, -10}

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 1148. 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 r = 1 r = 2 j = 1                       1 j = -1                     2   j = -3                   5 1   j = -5                 7 3     j = -7               9 4       j = -9             9 7         j = -11           9 9           j = -13         7 9             j = -15       5 9               j = -17     2 7                 j = -19   1 5                   j = -21   2                     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, Alternating, 48]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 48]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[14, 5, 15, 6], X[10, 8, 11, 7], > X[2, 9, 3, 10], X[16, 11, 17, 12], X[20, 13, 21, 14], X[6, 15, 7, 16], > X[22, 17, 1, 18], X[12, 19, 13, 20], X[18, 21, 19, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 48]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -4, 6, -10, 7, -3, 8, -6, 9, -11, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 48]]
Out[5]=	DTCode[4, 8, 14, 10, 2, 16, 20, 6, 22, 12, 18]
In[6]:=	alex = Alexander[Knot[11, Alternating, 48]][t]
Out[6]=	2 12 26 2 3 33 - -- + -- - -- - 26 t + 12 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 48]][z]
Out[7]=	2 6 1 + 4 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 48]}
In[9]:=	{KnotDet[Knot[11, Alternating, 48]], KnotSignature[Knot[11, Alternating, 48]]}
Out[9]=	{113, -4}
In[10]:=	J=Jones[Knot[11, Alternating, 48]][q]
Out[10]=	-11 3 7 12 16 18 18 16 11 7 3 1 - q + --- - -- + -- - -- + -- - -- + -- - -- + -- - - 10 9 8 7 6 5 4 3 2 q 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, Alternating, 48]}
In[12]:=	A2Invariant[Knot[11, Alternating, 48]][q]
Out[12]=	-34 -30 3 2 -24 2 3 3 2 -12 4 3 1 - q + q - --- + --- + q - --- + --- - --- + --- - q + --- - -- + 28 26 22 20 18 16 10 8 q q q q q q q q 2 -4 -2 > -- + q - q 6 q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 48]][a, z]
Out[13]=	2 8 10 2 2 4 2 6 2 8 2 10 2 2 4 4 4 a + a - a + 2 a z - a z + a z + 3 a z - a z + a z - 2 a z - 6 4 8 4 4 6 6 6 > a z + 2 a z - a z - a z
In[14]:=	Kauffman[Knot[11, Alternating, 48]][a, z]
Out[14]=	2 8 10 3 5 7 9 11 13 2 2 -a + a + a - a z - 3 a z - 2 a z - 2 a z - a z + a z + 3 a z + 4 2 6 2 8 2 10 2 12 2 3 3 5 3 > 2 a z - 4 a z - 10 a z - 5 a z + 2 a z + 6 a z + 11 a z + 7 3 9 3 11 3 13 3 2 4 6 4 8 4 > 7 a z + 7 a z + 3 a z - 2 a z - 3 a z + 16 a z + 26 a z + 10 4 12 4 3 5 5 5 7 5 9 5 11 5 > 8 a z - 5 a z - 8 a z - 14 a z - 5 a z - 7 a z - 7 a z + 13 5 2 6 4 6 6 6 8 6 10 6 12 6 > a z + a z - 7 a z - 23 a z - 27 a z - 9 a z + 3 a z + 3 7 5 7 7 7 11 7 4 8 6 8 8 8 > 3 a z + a z - 7 a z + 5 a z + 4 a z + 9 a z + 11 a z + 10 8 5 9 7 9 9 9 6 10 8 10 > 6 a z + 3 a z + 7 a z + 4 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 48]], Vassiliev[3][Knot[11, Alternating, 48]]}
Out[15]=	{4, -10}
In[16]:=	Kh[Knot[11, Alternating, 48]][q, t]
Out[16]=	3 5 1 2 1 5 2 7 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q q t q t q t q t q t q t q t 9 7 9 9 9 9 7 9 4 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 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 7 t 2 t 2 > ---- + -- + --- + q t 5 3 q q t q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a48

K11a47 K11a49