7 3

7_2

7_4

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7 3 Quick Notes

7 3 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X10,4,11,3 X14,8,1,7 X8,14,9,13 X12,6,13,5 X2,10,3,9 X4,12,5,11
Gauss code	1, -6, 2, -7, 5, -1, 3, -4, 6, -2, 7, -5, 4, -3
Dowker-Thistlethwaite code	6 10 12 14 2 4 8
Conway Notation	[43]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 2 Bridge index 2 Super bridge index [math]\displaystyle{ \{3,4\} }[/math] Nakanishi index 1 Maximal Thurston-Bennequin number [math]\displaystyle{ \text{$\$$Failed} }[/math] Hyperbolic Volume 4.59213 A-Polynomial See Data:7 3/A-polynomial

[edit Notes for 7 3's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus [math]\displaystyle{ 2 }[/math] Topological 4 genus [math]\displaystyle{ 2 }[/math] Concordance genus [math]\displaystyle{ \textrm{ConcordanceGenus}(\textrm{Knot}(7,3)) }[/math] Rasmussen s-Invariant -4

[edit Notes for 7 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial	[math]\displaystyle{ 2 t^2+2 t^{-2} -3 t-3 t^{-1} +3 }[/math]
Conway polynomial	[math]\displaystyle{ 2 z^4+5 z^2+1 }[/math]
2nd Alexander ideal (db, data sources)	[math]\displaystyle{ \{1\} }[/math]
Determinant and Signature	{ 13, 4 }
Jones polynomial	[math]\displaystyle{ -q^9+q^8-2 q^7+3 q^6-2 q^5+2 q^4-q^3+q^2 }[/math]
HOMFLY-PT polynomial (db, data sources)	[math]\displaystyle{ -z^2 a^{-8} -2 a^{-8} +z^4 a^{-6} +3 z^2 a^{-6} +2 a^{-6} +z^4 a^{-4} +3 z^2 a^{-4} + a^{-4} }[/math]
Kauffman polynomial (db, data sources)	[math]\displaystyle{ z^3 a^{-11} -2 z a^{-11} +z^4 a^{-10} -z^2 a^{-10} +z^5 a^{-9} -z^3 a^{-9} +z a^{-9} +z^6 a^{-8} -3 z^4 a^{-8} +6 z^2 a^{-8} -2 a^{-8} +2 z^5 a^{-7} -4 z^3 a^{-7} +3 z a^{-7} +z^6 a^{-6} -3 z^4 a^{-6} +4 z^2 a^{-6} -2 a^{-6} +z^5 a^{-5} -2 z^3 a^{-5} +z^4 a^{-4} -3 z^2 a^{-4} + a^{-4} }[/math]
The A2 invariant	[math]\displaystyle{ q^{-6} + q^{-10} + q^{-14} +2 q^{-16} + q^{-18} + q^{-20} - q^{-22} - q^{-24} - q^{-26} - q^{-28} }[/math]
The G2 invariant	[math]\displaystyle{ q^{-30} + q^{-34} - q^{-36} + q^{-38} + q^{-40} - q^{-42} +3 q^{-44} - q^{-46} +2 q^{-48} - q^{-52} +2 q^{-54} -2 q^{-56} +2 q^{-58} - q^{-62} +2 q^{-64} + q^{-68} +2 q^{-70} - q^{-72} +2 q^{-74} +3 q^{-80} -2 q^{-82} +3 q^{-84} +2 q^{-88} + q^{-90} -3 q^{-92} +3 q^{-94} -3 q^{-96} +2 q^{-98} - q^{-100} -3 q^{-102} + q^{-104} - q^{-106} - q^{-108} -3 q^{-112} - q^{-114} - q^{-116} -2 q^{-118} +2 q^{-120} -3 q^{-122} + q^{-124} - q^{-128} + q^{-130} - q^{-132} + q^{-134} - q^{-136} + q^{-138} - q^{-142} + q^{-144} + q^{-148} }[/math]

Further Quantum Invariants

Further quantum knot invariants for 7_3.

A1 Invariants.

Weight	Invariant
1	[math]\displaystyle{ q^{-3} + q^{-7} + q^{-11} + q^{-13} - q^{-15} - q^{-19} }[/math]
2	[math]\displaystyle{ q^{-6} +2 q^{-12} + q^{-14} - q^{-16} + q^{-18} + q^{-20} - q^{-22} + q^{-24} + q^{-26} - q^{-28} + q^{-34} -2 q^{-36} - q^{-38} + q^{-40} -2 q^{-42} - q^{-44} + q^{-46} + q^{-52} }[/math]
3	[math]\displaystyle{ q^{-9} + q^{-15} +2 q^{-17} + q^{-19} - q^{-21} - q^{-23} +2 q^{-25} +2 q^{-27} -2 q^{-31} +3 q^{-35} + q^{-37} -2 q^{-39} - q^{-41} +2 q^{-43} + q^{-45} -2 q^{-47} - q^{-49} + q^{-51} -2 q^{-55} - q^{-57} - q^{-59} +2 q^{-63} -2 q^{-65} -2 q^{-67} +3 q^{-71} - q^{-73} -3 q^{-75} +3 q^{-79} + q^{-81} - q^{-83} + q^{-87} + q^{-89} - q^{-99} }[/math]
4	[math]\displaystyle{ q^{-12} + q^{-18} + q^{-20} +2 q^{-22} - q^{-26} +4 q^{-32} +2 q^{-34} - q^{-36} -2 q^{-38} -3 q^{-40} +2 q^{-42} +3 q^{-44} +3 q^{-46} -5 q^{-50} - q^{-52} +2 q^{-54} +5 q^{-56} +2 q^{-58} -5 q^{-60} -4 q^{-62} - q^{-64} +4 q^{-66} +3 q^{-68} -4 q^{-70} -3 q^{-72} - q^{-74} +3 q^{-76} +2 q^{-78} -3 q^{-80} -2 q^{-82} - q^{-84} + q^{-86} -2 q^{-90} - q^{-92} - q^{-94} - q^{-96} + q^{-98} +4 q^{-100} - q^{-102} -2 q^{-104} -2 q^{-106} + q^{-108} +7 q^{-110} + q^{-112} -2 q^{-114} -5 q^{-116} - q^{-118} +7 q^{-120} +3 q^{-122} - q^{-124} -4 q^{-126} -3 q^{-128} +3 q^{-130} +2 q^{-132} +2 q^{-134} - q^{-136} -2 q^{-138} - q^{-142} + q^{-144} - q^{-148} - q^{-152} + q^{-160} }[/math]
5	[math]\displaystyle{ q^{-15} + q^{-21} + q^{-23} + q^{-25} + q^{-27} - q^{-31} +2 q^{-35} +2 q^{-37} +3 q^{-39} + q^{-41} -2 q^{-43} -4 q^{-45} -2 q^{-47} + q^{-49} +4 q^{-51} +5 q^{-53} +2 q^{-55} - q^{-57} -5 q^{-59} -5 q^{-61} +4 q^{-65} +7 q^{-67} +5 q^{-69} -2 q^{-71} -8 q^{-73} -8 q^{-75} - q^{-77} +6 q^{-79} +9 q^{-81} +4 q^{-83} -5 q^{-85} -11 q^{-87} -7 q^{-89} +3 q^{-91} +9 q^{-93} +7 q^{-95} -2 q^{-97} -9 q^{-99} -8 q^{-101} +7 q^{-105} +5 q^{-107} - q^{-109} -6 q^{-111} -5 q^{-113} +4 q^{-117} +3 q^{-119} - q^{-121} -3 q^{-123} - q^{-125} +2 q^{-129} + q^{-131} - q^{-133} - q^{-137} - q^{-139} - q^{-141} +2 q^{-143} +6 q^{-145} + q^{-147} - q^{-149} -3 q^{-151} -4 q^{-153} +2 q^{-155} +10 q^{-157} +7 q^{-159} -7 q^{-163} -11 q^{-165} -3 q^{-167} +8 q^{-169} +11 q^{-171} +4 q^{-173} -7 q^{-175} -12 q^{-177} -7 q^{-179} +3 q^{-181} +9 q^{-183} +7 q^{-185} + q^{-187} -6 q^{-189} -6 q^{-191} -3 q^{-193} +2 q^{-195} +4 q^{-197} +3 q^{-199} -2 q^{-203} -3 q^{-205} - q^{-207} + q^{-211} +2 q^{-213} - q^{-217} + q^{-225} + q^{-227} - q^{-235} }[/math]
6	[math]\displaystyle{ q^{-18} + q^{-24} + q^{-26} + q^{-28} + q^{-32} - q^{-36} + q^{-38} +2 q^{-40} +3 q^{-42} + q^{-44} +2 q^{-46} - q^{-48} -4 q^{-50} -3 q^{-52} - q^{-54} +3 q^{-56} +3 q^{-58} +7 q^{-60} +4 q^{-62} -2 q^{-64} -5 q^{-66} -6 q^{-68} -4 q^{-70} -3 q^{-72} +6 q^{-74} +9 q^{-76} +8 q^{-78} +3 q^{-80} -2 q^{-82} -8 q^{-84} -14 q^{-86} -5 q^{-88} +2 q^{-90} +10 q^{-92} +13 q^{-94} +10 q^{-96} - q^{-98} -16 q^{-100} -16 q^{-102} -12 q^{-104} + q^{-106} +13 q^{-108} +21 q^{-110} +12 q^{-112} -6 q^{-114} -16 q^{-116} -21 q^{-118} -10 q^{-120} +5 q^{-122} +21 q^{-124} +19 q^{-126} +2 q^{-128} -11 q^{-130} -21 q^{-132} -15 q^{-134} - q^{-136} +16 q^{-138} +17 q^{-140} +5 q^{-142} -5 q^{-144} -14 q^{-146} -11 q^{-148} -2 q^{-150} +11 q^{-152} +11 q^{-154} +2 q^{-156} -3 q^{-158} -8 q^{-160} -5 q^{-162} - q^{-164} +6 q^{-166} +5 q^{-168} -2 q^{-174} + q^{-178} +3 q^{-180} +2 q^{-182} -2 q^{-190} -2 q^{-192} -2 q^{-194} +2 q^{-196} +8 q^{-198} +3 q^{-200} +2 q^{-202} -3 q^{-204} -7 q^{-206} -8 q^{-208} - q^{-210} +14 q^{-212} +12 q^{-214} +9 q^{-216} -3 q^{-218} -16 q^{-220} -23 q^{-222} -11 q^{-224} +12 q^{-226} +19 q^{-228} +20 q^{-230} +4 q^{-232} -15 q^{-234} -28 q^{-236} -20 q^{-238} +4 q^{-240} +16 q^{-242} +24 q^{-244} +14 q^{-246} -2 q^{-248} -18 q^{-250} -20 q^{-252} -6 q^{-254} +3 q^{-256} +13 q^{-258} +14 q^{-260} +8 q^{-262} -4 q^{-264} -9 q^{-266} -7 q^{-268} -6 q^{-270} + q^{-272} +6 q^{-274} +6 q^{-276} + q^{-278} - q^{-280} - q^{-282} -4 q^{-284} -2 q^{-286} + q^{-288} +3 q^{-290} + q^{-292} + q^{-294} + q^{-296} -2 q^{-298} - q^{-300} + q^{-304} + q^{-310} - q^{-312} - q^{-314} - q^{-316} + q^{-324} }[/math]

A2 Invariants.

Weight	Invariant
1,0	[math]\displaystyle{ q^{-6} + q^{-10} + q^{-14} +2 q^{-16} + q^{-18} + q^{-20} - q^{-22} - q^{-24} - q^{-26} - q^{-28} }[/math]
1,1	[math]\displaystyle{ q^{-12} +2 q^{-16} -2 q^{-18} +6 q^{-20} +8 q^{-24} -2 q^{-26} +5 q^{-28} +2 q^{-34} -4 q^{-36} +6 q^{-38} -8 q^{-40} +4 q^{-42} -11 q^{-44} +4 q^{-46} -8 q^{-48} +4 q^{-50} - q^{-52} +4 q^{-56} -2 q^{-58} +3 q^{-60} -6 q^{-62} +2 q^{-64} -2 q^{-66} +2 q^{-68} -2 q^{-70} +2 q^{-72} + q^{-76} }[/math]
2,0	[math]\displaystyle{ q^{-12} + q^{-18} +2 q^{-20} + q^{-22} + q^{-24} +2 q^{-26} +2 q^{-28} + q^{-30} + q^{-32} + q^{-34} +2 q^{-40} - q^{-46} - q^{-48} -3 q^{-50} -3 q^{-52} -2 q^{-54} -2 q^{-56} -2 q^{-58} - q^{-60} + q^{-62} + q^{-64} +2 q^{-66} + q^{-68} + q^{-70} }[/math]
3,0	[math]\displaystyle{ q^{-18} +2 q^{-26} +3 q^{-28} +2 q^{-30} +2 q^{-36} +4 q^{-38} +2 q^{-40} +3 q^{-46} +4 q^{-48} +2 q^{-50} + q^{-54} +2 q^{-56} +2 q^{-58} - q^{-64} - q^{-66} -4 q^{-68} -4 q^{-70} -2 q^{-72} -3 q^{-74} -3 q^{-76} -5 q^{-78} -2 q^{-80} - q^{-82} -3 q^{-86} -3 q^{-88} - q^{-90} + q^{-92} -2 q^{-96} - q^{-98} +2 q^{-100} +4 q^{-102} +4 q^{-104} +3 q^{-106} +2 q^{-108} +3 q^{-110} +2 q^{-112} +2 q^{-114} - q^{-118} -2 q^{-120} -2 q^{-122} - q^{-124} - q^{-126} }[/math]

A3 Invariants.

Weight	Invariant
0,1,0	[math]\displaystyle{ q^{-12} + q^{-16} + q^{-18} +2 q^{-22} +3 q^{-24} + q^{-26} +3 q^{-28} +3 q^{-30} + q^{-32} -2 q^{-38} -2 q^{-40} -3 q^{-42} - q^{-44} -2 q^{-46} -2 q^{-48} + q^{-50} - q^{-52} - q^{-54} + q^{-56} + q^{-58} + q^{-62} }[/math]
1,0,0	[math]\displaystyle{ q^{-9} + q^{-13} + q^{-17} + q^{-19} +2 q^{-21} +2 q^{-23} + q^{-25} + q^{-27} - q^{-29} - q^{-31} -2 q^{-33} - q^{-35} - q^{-37} }[/math]
1,0,1	[math]\displaystyle{ q^{-18} +2 q^{-22} + q^{-26} +5 q^{-28} +2 q^{-30} +9 q^{-32} +5 q^{-34} +6 q^{-36} +8 q^{-38} +9 q^{-42} - q^{-44} + q^{-48} -8 q^{-50} -4 q^{-52} -7 q^{-54} -8 q^{-56} -7 q^{-58} -3 q^{-60} -6 q^{-62} - q^{-66} +7 q^{-70} -2 q^{-72} +7 q^{-74} +2 q^{-76} -3 q^{-78} +3 q^{-80} -4 q^{-82} - q^{-84} + q^{-86} -2 q^{-88} + q^{-90} - q^{-92} +2 q^{-96} + q^{-100} }[/math]

A4 Invariants.

Weight	Invariant
0,1,0,0	[math]\displaystyle{ q^{-18} + q^{-22} + q^{-24} + q^{-26} + q^{-28} +3 q^{-30} +2 q^{-32} +2 q^{-34} +4 q^{-36} +4 q^{-38} +3 q^{-40} +3 q^{-42} +4 q^{-44} +3 q^{-46} -2 q^{-52} -5 q^{-54} -6 q^{-56} -5 q^{-58} -6 q^{-60} -5 q^{-62} -2 q^{-64} - q^{-66} + q^{-70} +3 q^{-72} +2 q^{-74} + q^{-76} + q^{-78} + q^{-80} }[/math]
1,0,0,0	[math]\displaystyle{ q^{-12} + q^{-16} + q^{-20} + q^{-22} + q^{-24} +2 q^{-26} +2 q^{-28} +2 q^{-30} + q^{-32} + q^{-34} - q^{-36} - q^{-38} -2 q^{-40} -2 q^{-42} - q^{-44} - q^{-46} }[/math]

B2 Invariants.

Weight	Invariant
0,1	[math]\displaystyle{ q^{-12} + q^{-16} - q^{-18} +2 q^{-20} + q^{-24} + q^{-26} + q^{-28} + q^{-30} - q^{-32} +2 q^{-34} -2 q^{-36} +2 q^{-38} -2 q^{-40} + q^{-42} - q^{-44} - q^{-50} + q^{-52} - q^{-54} + q^{-56} - q^{-58} - q^{-62} }[/math]
1,0	[math]\displaystyle{ q^{-18} + q^{-26} + q^{-28} - q^{-32} + q^{-34} +2 q^{-36} +2 q^{-38} + q^{-42} + q^{-44} +2 q^{-46} + q^{-48} + q^{-54} - q^{-58} - q^{-60} - q^{-66} -2 q^{-68} - q^{-70} - q^{-74} -2 q^{-76} - q^{-78} + q^{-80} - q^{-84} - q^{-86} + q^{-90} + q^{-92} + q^{-100} }[/math]

D4 Invariants.

Weight	Invariant
1,0,0,0	[math]\displaystyle{ q^{-18} + q^{-22} +2 q^{-26} +2 q^{-30} + q^{-32} +2 q^{-34} +2 q^{-36} +2 q^{-38} +3 q^{-40} +3 q^{-42} +3 q^{-44} +2 q^{-48} -2 q^{-50} -4 q^{-54} -2 q^{-56} -4 q^{-58} - q^{-60} -2 q^{-62} - q^{-64} - q^{-66} - q^{-68} + q^{-70} - q^{-72} - q^{-76} + q^{-78} + q^{-82} + q^{-86} }[/math]

G2 Invariants.

Weight

Invariant

1,0

[math]\displaystyle{ q^{-30} + q^{-34} - q^{-36} + q^{-38} + q^{-40} - q^{-42} +3 q^{-44} - q^{-46} +2 q^{-48} - q^{-52} +2 q^{-54} -2 q^{-56} +2 q^{-58} - q^{-62} +2 q^{-64} + q^{-68} +2 q^{-70} - q^{-72} +2 q^{-74} +3 q^{-80} -2 q^{-82} +3 q^{-84} +2 q^{-88} + q^{-90} -3 q^{-92} +3 q^{-94} -3 q^{-96} +2 q^{-98} - q^{-100} -3 q^{-102} + q^{-104} - q^{-106} - q^{-108} -3 q^{-112} - q^{-114} - q^{-116} -2 q^{-118} +2 q^{-120} -3 q^{-122} + q^{-124} - q^{-128} + q^{-130} - q^{-132} + q^{-134} - q^{-136} + q^{-138} - q^{-142} + q^{-144} + q^{-148} }[/math]

.

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. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["7 3"];

In[4]:=

Alexander[K][t]

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

Out[4]=

[math]\displaystyle{ 2 t^2+2 t^{-2} -3 t-3 t^{-1} +3 }[/math]

In[5]:=

Conway[K][z]

Out[5]=

[math]\displaystyle{ 2 z^4+5 z^2+1 }[/math]

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]=

[math]\displaystyle{ \{1\} }[/math]

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 13, 4 }

In[8]:=

Jones[K][q]

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

Out[8]=

[math]\displaystyle{ -q^9+q^8-2 q^7+3 q^6-2 q^5+2 q^4-q^3+q^2 }[/math]

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

[math]\displaystyle{ -z^2 a^{-8} -2 a^{-8} +z^4 a^{-6} +3 z^2 a^{-6} +2 a^{-6} +z^4 a^{-4} +3 z^2 a^{-4} + a^{-4} }[/math]

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

[math]\displaystyle{ z^3 a^{-11} -2 z a^{-11} +z^4 a^{-10} -z^2 a^{-10} +z^5 a^{-9} -z^3 a^{-9} +z a^{-9} +z^6 a^{-8} -3 z^4 a^{-8} +6 z^2 a^{-8} -2 a^{-8} +2 z^5 a^{-7} -4 z^3 a^{-7} +3 z a^{-7} +z^6 a^{-6} -3 z^4 a^{-6} +4 z^2 a^{-6} -2 a^{-6} +z^5 a^{-5} -2 z^3 a^{-5} +z^4 a^{-4} -3 z^2 a^{-4} + a^{-4} }[/math]

Vassiliev invariants

V2 and V3:

(5, 11)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 [math]\displaystyle{ 20 }[/math] [math]\displaystyle{ 88 }[/math] [math]\displaystyle{ 200 }[/math] [math]\displaystyle{ \frac{1510}{3} }[/math] [math]\displaystyle{ \frac{242}{3} }[/math] [math]\displaystyle{ 1760 }[/math] [math]\displaystyle{ \frac{9520}{3} }[/math] [math]\displaystyle{ \frac{1696}{3} }[/math] [math]\displaystyle{ 440 }[/math] [math]\displaystyle{ \frac{4000}{3} }[/math] [math]\displaystyle{ 3872 }[/math] [math]\displaystyle{ \frac{30200}{3} }[/math] [math]\displaystyle{ \frac{4840}{3} }[/math] [math]\displaystyle{ \frac{121855}{6} }[/math] [math]\displaystyle{ 382 }[/math] [math]\displaystyle{ \frac{73862}{9} }[/math] [math]\displaystyle{ \frac{2437}{18} }[/math] [math]\displaystyle{ \frac{6559}{6} }[/math]

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 [math]\displaystyle{ t^rq^j }[/math] are shown, along with their alternating sums [math]\displaystyle{ \chi }[/math] (fixed [math]\displaystyle{ j }[/math], alternation over [math]\displaystyle{ r }[/math]). The squares with yellow highlighting are those on the "critical diagonals", where [math]\displaystyle{ j-2r=s+1 }[/math] or [math]\displaystyle{ j-2r=s+1 }[/math], where [math]\displaystyle{ s= }[/math]4 is the signature of 7 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \	0	1	2	3	4	5	6	7	χ
19								1	-1
17									0
15						2	1		-1
13					1				1
11				1	2				1
9			1	1					0
7			1						1
5	1	1							0
3	1								1

Computer Talk

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

[math]\displaystyle{ \textrm{Include}(\textrm{ColouredJonesM.mhtml}) }[/math]

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[7, 3]]
Out[2]=	7
In[3]:=	PD[Knot[7, 3]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 4, 11, 3], X[14, 8, 1, 7], X[8, 14, 9, 13], X[12, 6, 13, 5], X[2, 10, 3, 9], X[4, 12, 5, 11]]
In[4]:=	GaussCode[Knot[7, 3]]
Out[4]=	GaussCode[1, -6, 2, -7, 5, -1, 3, -4, 6, -2, 7, -5, 4, -3]
In[5]:=	BR[Knot[7, 3]]
Out[5]=	BR[3, {1, 1, 1, 1, 1, 2, -1, 2}]
In[6]:=	alex = Alexander[Knot[7, 3]][t]
Out[6]=	2 3 2 3 + -- - - - 3 t + 2 t 2 t t
In[7]:=	Conway[Knot[7, 3]][z]
Out[7]=	2 4 1 + 5 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[7, 3]}
In[9]:=	{KnotDet[Knot[7, 3]], KnotSignature[Knot[7, 3]]}
Out[9]=	{13, 4}
In[10]:=	J=Jones[Knot[7, 3]][q]
Out[10]=	2 3 4 5 6 7 8 9 q - q + 2 q - 2 q + 3 q - 2 q + q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[7, 3]}
In[12]:=	A2Invariant[Knot[7, 3]][q]
Out[12]=	6 10 14 16 18 20 22 24 26 28 q + q + q + 2 q + q + q - q - q - q - q
In[13]:=	Kauffman[Knot[7, 3]][a, z]
Out[13]=	2 2 2 2 3 3 -2 2 -4 2 z z 3 z z 6 z 4 z 3 z z z -- - -- + a - --- + -- + --- - --- + ---- + ---- - ---- + --- - -- - 8 6 11 9 7 10 8 6 4 11 9 a a a a a a a a a a a 3 3 4 4 4 4 5 5 5 6 6 4 z 2 z z 3 z 3 z z z 2 z z z z ---- - ---- + --- - ---- - ---- + -- + -- + ---- + -- + -- + -- 7 5 10 8 6 4 9 7 5 8 6 a a a a a a a a a a a
In[14]:=	{Vassiliev[2][Knot[7, 3]], Vassiliev[3][Knot[7, 3]]}
Out[14]=	{0, 11}
In[15]:=	Kh[Knot[7, 3]][q, t]
Out[15]=	3 5 5 7 2 9 2 9 3 11 3 11 4 13 4 q + q + q t + q t + q t + q t + q t + 2 q t + q t + 15 5 15 6 19 7 2 q t + q t + q t