10 124: Difference between revisions

Revision as of 20:05, 28 August 2005

Visit 10 124's page at the Knot Server (KnotPlot driven, includes 3D interactive images!)

Visit 10 124's page at the original Knot Atlas!

10_124 is also known as the torus knot T(5,3) or the pretzel knot P(5,3,-2). It is one of two knots which are both torus knots and pretzel knots, the other being 8_19 = T(4,3) = P(3,3,-2).

If one takes the symmetric diagram for 10_123 and makes it doubly alternating one gets a diagram for 10_124. That's the torus knot view. There is then a nice representation of the quandle of 10_124 into the dodecahedral quandle $Q_{30}$ . See [1].

10_124 is not $k$ -colourable for any $k$ . See The Determinant and the Signature.

Torus knot T(5,3) form

Knot presentations

Planar diagram presentation	X₄₂₅₁ X₈₄₉₃ X_9,17,10,16 X_5,15,6,14 X_15,7,16,6 X_11,19,12,18 X_13,1,14,20 X_17,11,18,10 X_19,13,20,12 X₂₈₃₇
Gauss code	1, -10, 2, -1, -4, 5, 10, -2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7
Dowker-Thistlethwaite code	4 8 -14 2 -16 -18 -20 -6 -10 -12
Conway Notation	[5,3,2-]

Three dimensional invariants

Symmetry type	Reversible
Unknotting number	4
3-genus	4
Bridge index	3
Super bridge index	Missing
Nakanishi index	1
Maximal Thurston-Bennequin number	[7][-15]
Hyperbolic Volume	Not hyperbolic
A-Polynomial	See Data:10 124/A-polynomial

[edit Notes for 10 124's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus	$4$
Topological 4 genus	$4$
Concordance genus	$4$
Rasmussen s-Invariant	-8

[edit Notes for 10 124's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-t^{3}+t-1+t^{-1}-t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+7z^{6}+14z^{4}+8z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 1, 8 }
Jones polynomial	$-q^{10}+q^{6}+q^{4}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-7z^{4}a^{-10}+21z^{2}a^{-8}-14z^{2}a^{-10}+z^{2}a^{-12}+7a^{-8}-8a^{-10}+2a^{-12}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-9}+z^{7}a^{-11}-8z^{6}a^{-8}-8z^{6}a^{-10}-7z^{5}a^{-9}-7z^{5}a^{-11}+21z^{4}a^{-8}+21z^{4}a^{-10}+14z^{3}a^{-9}+14z^{3}a^{-11}-21z^{2}a^{-8}-22z^{2}a^{-10}-z^{2}a^{-12}-8za^{-9}-8za^{-11}+7a^{-8}+8a^{-10}+2a^{-12}$
The A2 invariant	$q^{-14}+q^{-16}+2q^{-18}+2q^{-20}+2q^{-22}+q^{-24}-2q^{-28}-2q^{-30}-2q^{-32}-q^{-34}+q^{-40}$
The G2 invariant	$q^{-70}+q^{-72}+q^{-74}+q^{-76}+q^{-78}+2q^{-80}+2q^{-82}+q^{-84}+2q^{-86}+2q^{-88}+2q^{-90}+2q^{-92}+2q^{-94}+q^{-96}+2q^{-98}+q^{-100}+q^{-104}+q^{-106}-q^{-112}-q^{-114}-q^{-118}-2q^{-120}-2q^{-122}-q^{-124}-q^{-126}-2q^{-128}-2q^{-130}-2q^{-132}-q^{-134}-q^{-136}-2q^{-138}-2q^{-140}-q^{-142}-q^{-146}-q^{-148}+q^{-160}+q^{-162}+q^{-168}+q^{-170}+q^{-180}$

Further Quantum Invariants

Further quantum knot invariants for 10_124.

A1 Invariants.

Weight	Invariant
1	$q^{-7}+q^{-9}+q^{-11}+q^{-13}-q^{-19}-q^{-21}$
2	$q^{-14}+q^{-16}+q^{-18}+q^{-20}+q^{-22}+q^{-24}+q^{-26}-q^{-36}-q^{-38}-q^{-40}-q^{-42}-q^{-44}+q^{-60}$
3	$q^{-21}+q^{-23}+q^{-25}+q^{-27}+q^{-29}+q^{-31}+q^{-33}+q^{-35}+q^{-37}+q^{-39}-q^{-53}-q^{-55}-q^{-57}-q^{-59}-q^{-61}-q^{-63}-q^{-65}-q^{-67}+q^{-97}+q^{-99}+q^{-101}+q^{-103}-q^{-109}-q^{-111}$
5	$q^{-35}+q^{-37}+q^{-39}+q^{-41}+q^{-43}+q^{-45}+q^{-47}+q^{-49}+q^{-51}+q^{-53}+q^{-55}+q^{-57}+q^{-59}+q^{-61}+q^{-63}+q^{-65}-q^{-87}-q^{-89}-q^{-91}-q^{-93}-q^{-95}-q^{-97}-q^{-99}-q^{-101}-q^{-103}-q^{-105}-q^{-107}-q^{-109}-q^{-111}-q^{-113}+q^{-171}+q^{-173}+q^{-175}+q^{-177}+q^{-179}+q^{-181}+q^{-183}+q^{-185}+q^{-187}+q^{-189}-q^{-203}-q^{-205}-q^{-207}-q^{-209}-q^{-211}-q^{-213}-q^{-215}-q^{-217}+q^{-247}+q^{-249}+q^{-251}+q^{-253}-q^{-259}-q^{-261}$
6	$q^{-42}+q^{-44}+q^{-46}+q^{-48}+q^{-50}+q^{-52}+q^{-54}+q^{-56}+q^{-58}+q^{-60}+q^{-62}+q^{-64}+q^{-66}+q^{-68}+q^{-70}+q^{-72}+q^{-74}+q^{-76}+q^{-78}-q^{-104}-q^{-106}-q^{-108}-q^{-110}-q^{-112}-q^{-114}-q^{-116}-q^{-118}-q^{-120}-q^{-122}-q^{-124}-q^{-126}-q^{-128}-q^{-130}-q^{-132}-q^{-134}-q^{-136}+q^{-208}+q^{-210}+q^{-212}+q^{-214}+q^{-216}+q^{-218}+q^{-220}+q^{-222}+q^{-224}+q^{-226}+q^{-228}+q^{-230}+q^{-232}-q^{-250}-q^{-252}-q^{-254}-q^{-256}-q^{-258}-q^{-260}-q^{-262}-q^{-264}-q^{-266}-q^{-268}-q^{-270}+q^{-314}+q^{-316}+q^{-318}+q^{-320}+q^{-322}+q^{-324}+q^{-326}-q^{-336}-q^{-338}-q^{-340}-q^{-342}-q^{-344}+q^{-360}$

A2 Invariants.

Weight	Invariant
1,0	$q^{-14}+q^{-16}+2q^{-18}+2q^{-20}+2q^{-22}+q^{-24}-2q^{-28}-2q^{-30}-2q^{-32}-q^{-34}+q^{-40}$
1,1	$q^{-28}+2q^{-30}+4q^{-32}+6q^{-34}+7q^{-36}+8q^{-38}+6q^{-40}+4q^{-42}+q^{-44}-2q^{-46}-6q^{-48}-8q^{-50}-9q^{-52}-8q^{-54}-6q^{-56}-4q^{-58}-2q^{-60}+2q^{-64}+2q^{-66}+4q^{-68}+4q^{-70}+4q^{-72}+2q^{-74}+q^{-76}-2q^{-78}-2q^{-80}-2q^{-82}-q^{-84}+2q^{-90}$
2,0	$q^{-28}+q^{-30}+2q^{-32}+2q^{-34}+3q^{-36}+3q^{-38}+4q^{-40}+3q^{-42}+3q^{-44}+2q^{-46}+2q^{-48}-q^{-52}-3q^{-54}-4q^{-56}-5q^{-58}-5q^{-60}-5q^{-62}-4q^{-64}-3q^{-66}-2q^{-68}-q^{-70}+2q^{-74}+2q^{-76}+4q^{-78}+4q^{-80}+4q^{-82}+2q^{-84}+q^{-86}-2q^{-88}-2q^{-90}-2q^{-92}-q^{-94}+q^{-100}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{-28}+q^{-30}+3q^{-32}+4q^{-34}+6q^{-36}+6q^{-38}+7q^{-40}+3q^{-42}+q^{-44}-4q^{-46}-7q^{-48}-9q^{-50}-9q^{-52}-6q^{-54}-3q^{-56}+q^{-58}+3q^{-60}+4q^{-62}+2q^{-64}+2q^{-66}$
1,0,0	$q^{-21}+q^{-23}+2q^{-25}+3q^{-27}+3q^{-29}+3q^{-31}+2q^{-33}-2q^{-37}-3q^{-39}-4q^{-41}-3q^{-43}-2q^{-45}+q^{-49}+q^{-51}+q^{-53}$
1,0,1	$q^{-42}+2q^{-44}+5q^{-46}+9q^{-48}+14q^{-50}+19q^{-52}+23q^{-54}+22q^{-56}+19q^{-58}+10q^{-60}-q^{-62}-14q^{-64}-26q^{-66}-34q^{-68}-38q^{-70}-35q^{-72}-26q^{-74}-13q^{-76}-q^{-78}+11q^{-80}+18q^{-82}+21q^{-84}+19q^{-86}+14q^{-88}+8q^{-90}+4q^{-92}-3q^{-96}-3q^{-98}-4q^{-100}-3q^{-102}-3q^{-104}-q^{-106}-q^{-108}+2q^{-120}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-42}+q^{-44}+3q^{-46}+5q^{-48}+8q^{-50}+10q^{-52}+14q^{-54}+13q^{-56}+13q^{-58}+8q^{-60}+2q^{-62}-7q^{-64}-14q^{-66}-21q^{-68}-23q^{-70}-21q^{-72}-16q^{-74}-8q^{-76}-q^{-78}+7q^{-80}+9q^{-82}+11q^{-84}+9q^{-86}+7q^{-88}+3q^{-90}+2q^{-92}-q^{-96}-q^{-98}-q^{-100}-q^{-102}-q^{-104}$
1,0,0,0	$q^{-28}+q^{-30}+2q^{-32}+3q^{-34}+4q^{-36}+4q^{-38}+4q^{-40}+2q^{-42}+q^{-44}-2q^{-46}-4q^{-48}-5q^{-50}-5q^{-52}-4q^{-54}-2q^{-56}+q^{-60}+2q^{-62}+q^{-64}+q^{-66}$

B2 Invariants.

Weight	Invariant
0,1	$q^{-28}+q^{-30}+q^{-32}+2q^{-34}+2q^{-36}+2q^{-38}+q^{-40}+q^{-42}+q^{-44}-q^{-48}-q^{-50}-q^{-52}-2q^{-54}-q^{-56}-q^{-58}-q^{-60}$
1,0	$q^{-42}+q^{-46}+q^{-48}+2q^{-50}+q^{-52}+3q^{-54}+2q^{-56}+3q^{-58}+2q^{-60}+3q^{-62}+2q^{-64}+2q^{-66}-q^{-72}-2q^{-74}-3q^{-76}-3q^{-78}-3q^{-80}-4q^{-82}-3q^{-84}-3q^{-86}-2q^{-88}-2q^{-90}+q^{-96}+q^{-98}+2q^{-100}+q^{-102}+q^{-104}+q^{-106}+q^{-108}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-42}+q^{-44}+2q^{-46}+4q^{-48}+5q^{-50}+7q^{-52}+8q^{-54}+8q^{-56}+7q^{-58}+5q^{-60}+q^{-62}-3q^{-64}-7q^{-66}-10q^{-68}-11q^{-70}-11q^{-72}-9q^{-74}-6q^{-76}-2q^{-78}+q^{-80}+3q^{-82}+4q^{-84}+4q^{-86}+3q^{-88}+2q^{-90}+q^{-92}$

G2 Invariants.

Weight	Invariant
1,0	$q^{-70}+q^{-72}+q^{-74}+q^{-76}+q^{-78}+2q^{-80}+2q^{-82}+q^{-84}+2q^{-86}+2q^{-88}+2q^{-90}+2q^{-92}+2q^{-94}+q^{-96}+2q^{-98}+q^{-100}+q^{-104}+q^{-106}-q^{-112}-q^{-114}-q^{-118}-2q^{-120}-2q^{-122}-q^{-124}-q^{-126}-2q^{-128}-2q^{-130}-2q^{-132}-q^{-134}-q^{-136}-2q^{-138}-2q^{-140}-q^{-142}-q^{-146}-q^{-148}+q^{-160}+q^{-162}+q^{-168}+q^{-170}+q^{-180}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:= AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625.

In[3]:= K = Knot["10 124"];

In[4]:= Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]= $t^{4}-t^{3}+t-1+t^{-1}-t^{-3}+t^{-4}$

In[5]:= Conway[K][z]

Out[5]= $z^{8}+7z^{6}+14z^{4}+8z^{2}+1$

In[6]:= Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]= $\{1\}$

In[7]:= {KnotDet[K], KnotSignature[K]}

Out[7]= { 1, 8 }

In[8]:= Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]= $-q^{10}+q^{6}+q^{4}$

In[9]:= HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]= $z^{8}a^{-8}+8z^{6}a^{-8}-z^{6}a^{-10}+21z^{4}a^{-8}-7z^{4}a^{-10}+21z^{2}a^{-8}-14z^{2}a^{-10}+z^{2}a^{-12}+7a^{-8}-8a^{-10}+2a^{-12}$

In[10]:= Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]= $z^{8}a^{-8}+z^{8}a^{-10}+z^{7}a^{-9}+z^{7}a^{-11}-8z^{6}a^{-8}-8z^{6}a^{-10}-7z^{5}a^{-9}-7z^{5}a^{-11}+21z^{4}a^{-8}+21z^{4}a^{-10}+14z^{3}a^{-9}+14z^{3}a^{-11}-21z^{2}a^{-8}-22z^{2}a^{-10}-z^{2}a^{-12}-8za^{-9}-8za^{-11}+7a^{-8}+8a^{-10}+2a^{-12}$

Vassiliev invariants

V₂ and V₃:

(8, 20)

V_2,1 through V_6,9:

V_2,1

V_3,1

V_4,1

V_4,2

V_4,3

V_5,1

V_5,2

V_5,3

V_5,4

V_6,1

V_6,2

V_6,3

V_6,4

V_6,5

V_6,6

V_6,7

V_6,8

V_6,9

32

160

512

{\frac {3184}{3}}

{\frac {416}{3}}

5120

{\frac {23680}{3}}

{\frac {4000}{3}}

832

{\frac {16384}{3}}

12800

{\frac {101888}{3}}

{\frac {13312}{3}}

{\frac {910684}{15}}

{\frac {59264}{15}}

{\frac {869536}{45}}

{\frac {2756}{9}}

{\frac {34684}{15}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials

t^{r}q^{j}

are shown, along with their alternating sums

\chi

(fixed

j

, alternation over

r

). The squares with yellow highlighting are those on the "critical diagonals", where

j-2r=s+1

or

j-2r=s-1

, where

s=

8 is the signature of 10 124. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\		r
	\
j		\

0

1

2

3

4

5

6

7

χ

21

1

-1

19

1

-1

17

1

0

15

1

0

13

1

11

1

9

1

7

1

Integral Khovanov Homology

(db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$	$i=5$	$i=7$	$i=9$
$r=0$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=1$
$r=2$		${\mathbb {Z} }$
$r=3$		${\mathbb {Z} }_{2}$	${\mathbb {Z} }$
$r=4$	${\mathbb {Z} }$	${\mathbb {Z} }$
$r=5$		${\mathbb {Z} }$	${\mathbb {Z} }$
$r=6$	${\mathbb {Z} }$
$r=7$	${\mathbb {Z} }_{2}$	${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 124]]
Out[2]=	10
In[3]:=	PD[Knot[10, 124]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[9, 17, 10, 16], X[5, 15, 6, 14], X[15, 7, 16, 6], X[11, 19, 12, 18], X[13, 1, 14, 20], X[17, 11, 18, 10], X[19, 13, 20, 12], X[2, 8, 3, 7]]
In[4]:=	GaussCode[Knot[10, 124]]
Out[4]=	GaussCode[1, -10, 2, -1, -4, 5, 10, -2, -3, 8, -6, 9, -7, 4, -5, 3, -8, 6, -9, 7]
In[5]:=	BR[Knot[10, 124]]
Out[5]=	BR[3, {1, 1, 1, 1, 1, 2, 1, 1, 1, 2}]
In[6]:=	alex = Alexander[Knot[10, 124]][t]
Out[6]=	-4 -3 1 3 4 -1 + t - t + - + t - t + t t
In[7]:=	Conway[Knot[10, 124]][z]
Out[7]=	2 4 6 8 1 + 8 z + 14 z + 7 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 124]}
In[9]:=	{KnotDet[Knot[10, 124]], KnotSignature[Knot[10, 124]]}
Out[9]=	{1, 8}
In[10]:=	J=Jones[Knot[10, 124]][q]
Out[10]=	4 6 10 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] \|\| (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 124]}
In[12]:=	A2Invariant[Knot[10, 124]][q]
Out[12]=	14 16 18 20 22 24 28 30 32 34 q + q + 2 q + 2 q + 2 q + q - 2 q - 2 q - 2 q - q + 40 q
In[13]:=	Kauffman[Knot[10, 124]][a, z]
Out[13]=	2 2 2 3 3 2 8 7 8 z 8 z z 22 z 21 z 14 z 14 z --- + --- + -- - --- - --- - --- - ----- - ----- + ----- + ----- + 12 10 8 11 9 12 10 8 11 9 a a a a a a a a a a 4 4 5 5 6 6 7 7 8 8 21 z 21 z 7 z 7 z 8 z 8 z z z z z ----- + ----- - ---- - ---- - ---- - ---- + --- + -- + --- + -- 10 8 11 9 10 8 11 9 10 8 a a a a a a a a a a
In[14]:=	{Vassiliev[2][Knot[10, 124]], Vassiliev[3][Knot[10, 124]]}
Out[14]=	{0, 20}
In[15]:=	Kh[Knot[10, 124]][q, t]
Out[15]=	7 9 11 2 15 3 13 4 15 4 17 5 19 5 q + q + q t + q t + q t + q t + q t + q t + 17 6 21 7 q t + q t

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+<!-- -->
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 {{Knot Navigation Links|ext=gif}}
+{{Rolfsen Knot Page Header|n=10|k=124|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-1,-4,5,10,-2,-3,8,-6,9,-7,4,-5,3,-8,6,-9,7/goTop.html}}
-{| align=left
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-|{{Rolfsen Knot Site Links|n=10|k=124|KnotilusURL=http://srankin.math.uwo.ca/cgi-bin/retrieve.cgi/1,-10,2,-1,-4,5,10,-2,-3,8,-6,9,-7,4,-5,3,-8,6,-9,7/goTop.html}}
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 {{Vassiliev Invariants}}
-===[[Khovanov Homology]]===
+{{Khovanov Homology|table=<table border=1>
-The coefficients of the monomials <math>t^rq^j</math> are shown, along with their alternating sums <math>\chi</math> (fixed <math>j</math>, alternation over <math>r</math>). The squares with <font class=HLYellow>yellow</font> highlighting are those on the "critical diagonals", where <math>j-2r=s+1</math> or <math>j-2r=s+1</math>, where <math>s=</math>{{Data:{{PAGENAME}}/Signature}} is the signature of {{PAGENAME}}. Nonzero entries off the critical diagonals (if any exist) are highlighted in <font class=HLRed>red</font>.
-<center><table border=1>
 <tr align=center>
 <td width=16.6667%><table cellpadding=0 cellspacing=0>
@@ Line 44: / Line 37: @@
 <tr align=center><td>9</td><td bgcolor=yellow>1</td><td bgcolor=yellow>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
 <tr align=center><td>7</td><td bgcolor=yellow>1</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>&nbsp;</td><td>1</td></tr>
-</table></center>
+</table>}}
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   q   t  + q   t</nowiki></pre></td></tr>
 </table>
+ [[Category:Knot Page]]

10 124: Difference between revisions

Revision as of 20:05, 28 August 2005

Contents

Knot presentations

Three dimensional invariants

Four dimensional invariants

Polynomial invariants

Vassiliev invariants

Khovanov Homology

Computer Talk

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